Merge
authorGraham Klyne
Fri, 05 Apr 2013 18:13:06 +0100
changeset 6062 0c673a123458
parent 6061 04c325fb6e36 (current diff)
parent 6058 ff8007656fc6 (diff)
child 6063 2072b00f4563
Merge
primer/NOTE-prov-primer-20130430/primer-provn-examples.txt
primer/NOTE-prov-primer-20130430/primer-turtle-examples.txt
primer/primer-provn-examples.txt
primer/primer-turtle-examples.txt
Binary file primer/NOTE-prov-primer-20130430/images/everything.png has changed
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/primer/NOTE-prov-primer-20130430/primer-provn-examples.provn	Fri Apr 05 18:13:06 2013 +0100
@@ -0,0 +1,84 @@
+prefix prov <http://www.w3.org/ns/prov#>
+prefix ex <http://www.example.org#>
+prefix dcterms <http://purl.org/dc/terms/>
+prefix foaf <http://xmlns.com/foaf/0.1/>
+
+// The examples below are taken from, and follow the same order
+// as in, the W3C PROV Primer, http://www.w3.org/TR/prov-primer/
+
+// Entities
+
+entity(exn:article, [dcterms:title="Crime rises in cities"])
+entity(exg:dataset1)
+entity(exc:regionList)
+entity(exc:composition1)
+entity(exc:chart1)
+    
+// Activities
+
+activity(exc:compile1)
+activity(exc:compose1)
+activity(exc:illustrate1)
+    
+// Usage and Generation
+
+used(exc:compose1, exg:dataset1, -)
+used(exc:compose1, exc:regionList, -)
+wasGeneratedBy(exc:composition1, exc:compose1, -)
+used(exc:illustrate1, exc:composition1, -)
+wasGeneratedBy(exc:chart1, exc:illustrate1, -)
+    
+// Agents and Responsibility
+
+wasAssociatedWith(exc:compose1, exc:derek, -)
+wasAssociatedWith(exc:illustrate1, exc:derek, -)
+agent(exc:derek,
+      [prov:type='prov:Person', foaf:givenName="Derek", 
+       foaf:mbox="<mailto:[email protected]>"])
+
+agent(exc:chartgen,
+      [prov:type='prov:Organization',
+       foaf:name="Chart Generators Inc"])
+actedOnBehalfOf(exc:derek, exc:chartgen)
+
+wasAttributedTo(exc:chart1, exc:derek)
+    
+// Roles
+
+used(exc:compose1, exg:dataset1, -, [prov:role='exc:dataToCompose'])
+used(exc:compose1, exc:regionList, -, [prov:role='exc:regionsToAggregateBy'])
+wasAssociatedWith(exc:compose1, exc:derek, -, [prov:role='exc:analyst'])
+wasGeneratedBy(exc:composition1, exc:compose1, -, [prov:role='exc:composedData'])
+
+// Derivation and Revision
+
+entity(exg:dataSet2)
+wasDerivedFrom(exg:dataSet2, exg:dataset1, [prov:type='prov:Revision'])
+wasDerivedFrom(exc:chart2, exg:dataSet2)
+entity(exc:chart2)
+wasDerivedFrom(exc:chart2, exc:chart1, [prov:type='prov:Revision'])
+    
+// Plans
+
+activity(exg:correct1)
+agent(exg:edith, [prov:type='prov:Person'])
+entity(exg:instructions)
+wasAssociatedWith(exg:correct1, exg:edith, exg:instructions)
+wasGeneratedBy(exg:dataSet2, exg:correct1, -)
+    
+// Time
+
+wasGeneratedBy(exc:chart1, exc:compile1,  2012-03-02T10:30:00)
+wasGeneratedBy(exc:chart2, exc:compile2, 2012-04-01T15:21:00)
+activity(exg:correct1, 2012-03-31T09:21:00, 2012-04-01T15:21:00)
+    
+// Alternate Entities and Specialization
+
+entity(ex:quoteInBlogEntry-20130326)
+wasDerivedFrom(ex:quoteInBlogEntry-20130326, exn:article, [prov:type='prov:Quotation'])
+    
+entity(exn:articleV1)
+specializationOf(exn:articleV1, exn:article)    
+specializationOf(exn:articleV2, exn:article)
+alternateOf(exn:articleV2, exn:articleV1)
+    
\ No newline at end of file
--- a/primer/NOTE-prov-primer-20130430/primer-provn-examples.txt	Fri Apr 05 18:11:45 2013 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,84 +0,0 @@
-prefix prov <http://www.w3.org/ns/prov#>
-prefix ex <http://www.example.org#>
-prefix dcterms <http://purl.org/dc/terms/>
-prefix foaf <http://xmlns.com/foaf/0.1/>
-
-// The examples below are taken from, and follow the same order
-// as in, the W3C PROV Primer, http://www.w3.org/TR/prov-primer/
-
-// Entities
-
-entity(exn:article, [dcterms:title="Crime rises in cities"])
-entity(exg:dataset1)
-entity(exc:regionList)
-entity(exc:composition1)
-entity(exc:chart1)
-    
-// Activities
-
-activity(exc:compile1)
-activity(exc:compose1)
-activity(exc:illustrate1)
-    
-// Usage and Generation
-
-used(exc:compose1, exg:dataset1, -)
-used(exc:compose1, exc:regionList, -)
-wasGeneratedBy(exc:composition1, exc:compose1, -)
-used(exc:illustrate1, exc:composition1, -)
-wasGeneratedBy(exc:chart1, exc:illustrate1, -)
-    
-// Agents and Responsibility
-
-wasAssociatedWith(exc:compose1, exc:derek, -)
-wasAssociatedWith(exc:illustrate1, exc:derek, -)
-agent(exc:derek,
-      [prov:type='prov:Person', foaf:givenName="Derek", 
-       foaf:mbox="<mailto:[email protected]>"])
-
-agent(exc:chartgen,
-      [prov:type='prov:Organization',
-       foaf:name="Chart Generators Inc"])
-actedOnBehalfOf(exc:derek, exc:chartgen)
-
-wasAttributedTo(exc:chart1, exc:derek)
-    
-// Roles
-
-used(exc:compose1, exg:dataset1, -, [prov:role='exc:dataToCompose'])
-used(exc:compose1, exc:regionList, -, [prov:role='exc:regionsToAggregateBy'])
-wasAssociatedWith(exc:compose1, exc:derek, -, [prov:role='exc:analyst'])
-wasGeneratedBy(exc:composition1, exc:compose1, -, [prov:role='exc:composedData'])
-
-// Derivation and Revision
-
-entity(exg:dataSet2)
-wasDerivedFrom(exg:dataSet2, exg:dataset1, [prov:type='prov:Revision'])
-wasDerivedFrom(exc:chart2, exg:dataSet2)
-entity(exc:chart2)
-wasDerivedFrom(exc:chart2, exc:chart1, [prov:type='prov:Revision'])
-    
-// Plans
-
-activity(exg:correct1)
-agent(exg:edith, [prov:type='prov:Person'])
-entity(exg:instructions)
-wasAssociatedWith(exg:correct1, exg:edith, exg:instructions)
-wasGeneratedBy(exg:dataSet2, exg:correct1, -)
-    
-// Time
-
-wasGeneratedBy(exc:chart1, exc:compile1,  2012-03-02T10:30:00)
-wasGeneratedBy(exc:chart2, exc:compile2, 2012-04-01T15:21:00)
-activity(exg:correct1, 2012-03-31T09:21:00, 2012-04-01T15:21:00)
-    
-// Alternate Entities and Specialization
-
-entity(ex:quoteInBlogEntry-20130326)
-wasDerivedFrom(ex:quoteInBlogEntry-20130326, exn:article, [prov:type='prov:Quotation'])
-    
-entity(exn:articleV1)
-specializationOf(exn:articleV1, exn:article)    
-specializationOf(exn:articleV2, exn:article)
-alternateOf(exn:articleV2, exn:articleV1)
-    
\ No newline at end of file
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/primer/NOTE-prov-primer-20130430/primer-turtle-examples.ttl	Fri Apr 05 18:13:06 2013 +0100
@@ -0,0 +1,116 @@
[email protected] rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#> .
[email protected] prov: <http://www.w3.org/ns/prov#> .
[email protected] exn: <http://www.example.org#> .
[email protected] exg: <http://www.example.org#> .
[email protected] exc: <http://www.example.org#> .
[email protected] exb: <http://www.example.org#> .
[email protected] dcterms: <http://purl.org/dc/terms/> .
[email protected] foaf: <http://xmlns.com/foaf/0.1/> .
+
+# The examples below are taken from, and follow the same order
+# as in the W3C PROV Primer, http://www.w3.org/TR/prov-primer/
+
+# Entities
+
+exn:article      a prov:Entity ;
+                 dcterms:title "Crime rises in cities" .
+exg:dataset1     a prov:Entity .
+exc:regionList   a prov:Entity .
+exc:composition1 a prov:Entity .
+exc:chart1       a prov:Entity .
+
+# Activities
+
+exc:compile1    a prov:Activity .
+exc:compose1    a prov:Activity .
+exc:illustrate1 a prov:Activity .
+
+# Usage and Generation
+
+exc:compose1     prov:used           exg:dataset1 ;
+                 prov:used           exc:regionList .
+exc:composition1 prov:wasGeneratedBy exc:compose1 .
+exc:illustrate1  prov:used           exc:composition1 .
+exc:chart1       prov:wasGeneratedBy exc:illustrate1 .
+
+# Agents and Responsibility
+
+exc:compose1    prov:wasAssociatedWith exc:derek .
+exc:illustrate1 prov:wasAssociatedWith exc:derek .
+exc:derek a prov:Agent ;
+          a prov:Person ;
+          foaf:givenName "Derek"^^xsd:string ;
+          foaf:mbox      <mailto:[email protected]> .
+exc:derek prov:actedOnBehalfOf exc:chartgen .
+exc:chartgen a prov:Agent ;
+             a prov:Organization ;
+             foaf:name "Chart Generators Inc" .
+exc:chart1 prov:wasAttributedTo exc:derek .
+
+# Roles
+    
+exc:dataToCompose        a prov:Role .
+exc:regionsToAggregateBy a prov:Role .
+exc:composedData         a prov:Role .
+exc:analyst              a prov:Role .
+exc:compose1 prov:qualifiedUsage [
+               a prov:Usage ;
+               prov:entity  exg:dataset1 ;
+               prov:hadRole exc:dataToCompose 
+] .    
+exc:compose1 prov:qualifiedUsage [
+               a prov:Usage ;
+               prov:entity  exc:regionList ;
+               prov:hadRole exc:regionsToAggregateBy
+] .
+exc:compose1 prov:qualifiedAssociation [
+               a prov:Association ;
+               prov:agent    exc:derek ;
+               prov:hadRole  exc:analyst
+] .
+exc:composition1 prov:qualifiedGeneration [
+                   a prov:Generation ;
+                   prov:activity  exc:compose1 ;
+                   prov:hadRole   exc:composedData
+] .
+
+# Revision and Derivation
+
+exg:dataSet2 a prov:Entity ;
+             prov:wasRevisionOf exg:dataset1 .
+exc:chart2   a prov:Entity ;
+             prov:wasDerivedFrom exg:dataSet2 .
+exc:chart2   a prov:Entity ;
+             prov:wasRevisionOf exc:chart1 .
+    
+# Plans
+
+exg:correct1    a prov:Activity .
+exg:edith       a prov:Agent, prov:Person .
+exg:instructions a prov:Plan .
+    
+exg:correct1 prov:qualifiedAssociation [
+               a Association ;
+               prov:agent   exg:edith ;
+               prov:hadPlan exg:instructions
+] .
+exg:dataSet2 prov:wasGeneratedBy exg:correct1 .
+    
+# Time
+
+exc:chart1 prov:generatedAtTime "2012-03-02T10:30:00"^^xsd:dateTime .
+exc:chart2 prov:generatedAtTime "2012-04-01T15:21:00"^^xsd:dateTime .
+    
+exg:correct1 prov:startedAtTime "2012-03-31T09:21:00"^^xsd:dateTime ;
+             prov:endedAtTime   "2012-04-01T15:21:00"^^xsd:dateTime .
+    
+# Alternate Entities and Specialization
+
+ex:quoteInBlogEntry-20130326 a prov:Entity ;
+                             prov:wasQuotedFrom exn:article .
+
+exn:articleV1 a prov:Entity ;
+              prov:specializationOf exn:article .
+    
+exn:articleV2 prov:specializationOf exn:article .
+exn:articleV2 prov:alternateOf      exn:articleV1 .
\ No newline at end of file
--- a/primer/NOTE-prov-primer-20130430/primer-turtle-examples.txt	Fri Apr 05 18:11:45 2013 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,116 +0,0 @@
[email protected] rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#> .
[email protected] prov: <http://www.w3.org/ns/prov#> .
[email protected] exn: <http://www.example.org#> .
[email protected] exg: <http://www.example.org#> .
[email protected] exc: <http://www.example.org#> .
[email protected] exb: <http://www.example.org#> .
[email protected] dcterms: <http://purl.org/dc/terms/> .
[email protected] foaf: <http://xmlns.com/foaf/0.1/> .
-
-# The examples below are taken from, and follow the same order
-# as in the W3C PROV Primer, http://www.w3.org/TR/prov-primer/
-
-# Entities
-
-exn:article      a prov:Entity ;
-                 dcterms:title "Crime rises in cities" .
-exg:dataset1     a prov:Entity .
-exc:regionList   a prov:Entity .
-exc:composition1 a prov:Entity .
-exc:chart1       a prov:Entity .
-
-# Activities
-
-exc:compile1    a prov:Activity .
-exc:compose1    a prov:Activity .
-exc:illustrate1 a prov:Activity .
-
-# Usage and Generation
-
-exc:compose1     prov:used           exg:dataset1 ;
-                 prov:used           exc:regionList .
-exc:composition1 prov:wasGeneratedBy exc:compose1 .
-exc:illustrate1  prov:used           exc:composition1 .
-exc:chart1       prov:wasGeneratedBy exc:illustrate1 .
-
-# Agents and Responsibility
-
-exc:compose1    prov:wasAssociatedWith exc:derek .
-exc:illustrate1 prov:wasAssociatedWith exc:derek .
-exc:derek a prov:Agent ;
-          a prov:Person ;
-          foaf:givenName "Derek"^^xsd:string ;
-          foaf:mbox      <mailto:[email protected]> .
-exc:derek prov:actedOnBehalfOf exc:chartgen .
-exc:chartgen a prov:Agent ;
-             a prov:Organization ;
-             foaf:name "Chart Generators Inc" .
-exc:chart1 prov:wasAttributedTo exc:derek .
-
-# Roles
-    
-exc:dataToCompose        a prov:Role .
-exc:regionsToAggregateBy a prov:Role .
-exc:composedData         a prov:Role .
-exc:analyst              a prov:Role .
-exc:compose1 prov:qualifiedUsage [
-               a prov:Usage ;
-               prov:entity  exg:dataset1 ;
-               prov:hadRole exc:dataToCompose 
-] .    
-exc:compose1 prov:qualifiedUsage [
-               a prov:Usage ;
-               prov:entity  exc:regionList ;
-               prov:hadRole exc:regionsToAggregateBy
-] .
-exc:compose1 prov:qualifiedAssociation [
-               a prov:Association ;
-               prov:agent    exc:derek ;
-               prov:hadRole  exc:analyst
-] .
-exc:composition1 prov:qualifiedGeneration [
-                   a prov:Generation ;
-                   prov:activity  exc:compose1 ;
-                   prov:hadRole   exc:composedData
-] .
-
-# Revision and Derivation
-
-exg:dataSet2 a prov:Entity ;
-             prov:wasRevisionOf exg:dataset1 .
-exc:chart2   a prov:Entity ;
-             prov:wasDerivedFrom exg:dataSet2 .
-exc:chart2   a prov:Entity ;
-             prov:wasRevisionOf exc:chart1 .
-    
-# Plans
-
-exg:correct1    a prov:Activity .
-exg:edith       a prov:Agent, prov:Person .
-exg:instructions a prov:Plan .
-    
-exg:correct1 prov:qualifiedAssociation [
-               a Association ;
-               prov:agent   exg:edith ;
-               prov:hadPlan exg:instructions
-] .
-exg:dataSet2 prov:wasGeneratedBy exg:correct1 .
-    
-# Time
-
-exc:chart1 prov:generatedAtTime "2012-03-02T10:30:00"^^xsd:dateTime .
-exc:chart2 prov:generatedAtTime "2012-04-01T15:21:00"^^xsd:dateTime .
-    
-exg:correct1 prov:startedAtTime "2012-03-31T09:21:00"^^xsd:dateTime ;
-             prov:endedAtTime   "2012-04-01T15:21:00"^^xsd:dateTime .
-    
-# Alternate Entities and Specialization
-
-ex:quoteInBlogEntry-20130326 a prov:Entity ;
-                             prov:wasQuotedFrom exn:article .
-
-exn:articleV1 a prov:Entity ;
-              prov:specializationOf exn:article .
-    
-exn:articleV2 prov:specializationOf exn:article .
-exn:articleV2 prov:alternateOf      exn:articleV1 .
\ No newline at end of file
--- a/primer/Primer.html	Fri Apr 05 18:11:45 2013 +0100
+++ b/primer/Primer.html	Fri Apr 05 18:13:06 2013 +0100
@@ -112,7 +112,7 @@
   <section id="abstract">
    <p>
     This document provides an intuitive introduction and guide to the
-    PROV specification for provenance on the Web. PROV is a core data model for
+    PROV Data Model for provenance interchange on the web. PROV defines a core data model for
     provenance for building representations of the entities, people and
     processes involved in producing a piece of data or thing in the world.
     This primer explains the fundamental PROV concepts and provides examples 
@@ -162,9 +162,9 @@
    <h2>Introduction</h2>
    <p>
     This primer document provides an accessible introduction to the PROV 
-    specification for provenance on the Web.  
+    Data Model for provenance interchange on the web.
     The <i>provenance</i> of digital objects represents their origins.  PROV is a 
-    proposed specification to express provenance records, 
+    specification to express provenance records, 
     which contain <i>descriptions</i> of the entities 
     and activities involved in producing and delivering or otherwise influencing a 
     given object.
@@ -172,7 +172,7 @@
     to records of provenance, except where the distinction is important for clarity.
     Provenance can be used for many purposes, such as 
     understanding how data was collected so it can be meaningfully used, determining 
-    ownership and rights over an object, making judgments about information to 
+    ownership and rights over an object, making judgements about information to 
     determine whether to trust it, verifying that the process and steps used to obtain a 
     result complies with given requirements, and reproducing how something was generated.
    </p>
@@ -183,7 +183,7 @@
     and as a result different types of information might be captured in provenance records.
    <ul>
     <li>
-     One perspective might focus on <i>agent-centered provenance</i>, that is, what entities 
+     One perspective might focus on <i>agent-centered provenance</i>, that is, what people or organizations 
      were involved in generating or manipulating the information in question.  For example, 
      in the provenance of a picture in a news article we might capture the photographer who 
      took it, the person that edited it, and the newspaper that published it.
@@ -233,7 +233,7 @@
 
   <p>There are additional reference documents for PROV that are not covered in this 
    primer, including the PROV Access and Query aspects of the specification [[PROV-AQ]], 
-   the constraints on the PROV data model [[PROV-CONSTRAINTS]], 
+   the constraints on the PROV data model [[PROV-CONSTRAINTS]], and
    a formal semantics of the PROV data model [[PROV-SEM]]. </p>
 
  </section>
@@ -410,7 +410,7 @@
     the same thing together through the mechanism of <i>specialization</i>. One
     entity is a specialization of another entity if it shares the same fixed attributes,
     with the possible addition of further fixed attributes. This concept is best
-    illustrated through a few use cases, as below.
+    illustrated through a few use cases.
    </p>
    <p>
     Entities can be mutable things. For example, a webpage is a single
@@ -425,9 +425,9 @@
     to themselves using PROV while, independently, a reader might quote a fact from that
     article elsewhere and document this in PROV. If the author later changes the fact,
     then from the reader's perspective there are now two versions of the article, and
-    they had quoted from the version before the change. From the author's perspective,
-    there is a single article, attributed to the author. If either of them, or a
-    third party, were to connect the two PROV records, they would say that the
+    the reader had quoted from the version before the change. From the author's perspective,
+    there is a single article, attributed to the author. If the author, the reader, or a
+    third party, were to connect the two PROV records, that party would say that the
     article as referred to by the reader is a specialization of the same article
     as referred to by the author.
    </p>
@@ -609,7 +609,7 @@
     vocabulary.
    </p>
    <p>
-    PROV data is commonly visualized for human consumption using particular conventions,
+    PROV data is commonly visualized for human consumption using <a href="http://www.w3.org/2011/prov/wiki/Diagrams">particular conventions</a>,
     which we will introduce over the following sections. To start with, entities
     are denoted using ovals, as shown below.
    </p>
@@ -1965,7 +1965,7 @@
   </section>
 
   <section>
-   <h3>Additional aspects of PROV</h3>
+   <h3>Additional Aspects of PROV</h3>
    <p>There are additional aspects of the PROV data model that we have not covered 
     in this primer and that are important in many uses of provenance. These include
     the following.
@@ -1986,179 +1986,183 @@
      He would describe all the charts as a collection, and then state the provenance of 
      the collection to be a single step that generates all the charts. </li>
    </ol>
-     <p>
-      Please see the [[PROV-DM]] specification for details on these
+   <p>
+    Please see the [[PROV-DM]] specification for details on these
     and other concepts.
-     </p>
-     </section>
-
-     <section>
-      <h3>Summary</h3>
-      <p>
-       The full set of examples shown above are available in <a href="primer-turtle-examples.txt">PROV-O Turtle format</a>
-       or <a href="primer-provn-examples.txt">PROV-N format</a> or <a href="primer-prov-xml-examples.xml">PROV-XML format</a>.
-       We visualize the whole example below.
-      </p>
-      <img src="images/everything.png" alt="Provenance graph for whole example"/>
-      <p>These examples highlight the following important capabilities of PROV:
-      </p>
-      <ol>
-       <li><b>Representing diverse entities involved in provenance</b>.  
-        The examples illustrate that provenance records can contain different 
-        kinds of information:
-        <ul>
-         <li>The newspaper took an object-centered view on their provenance statements, 
-          stating what objects were derived from others but not stating what processes 
-          were used or what entities were involved.</li>
-         <li>Betty the blogger took a very agent-centered view on provenance.  
-          She chose to associate her blog post with herself, the government agency 
-          that published the data, and the newspaper that published the article.</li>
-         <li>Derek took a process-centered perspective and combined it with object-centered 
-          and entity-centered provenance information. The processes followed to derive the 
-          charts from the government datasets were clearly.  All intermediate and final 
-          documents were stated.  His involvement in the various activities was clearly 
-          recorded, and the resulting documents were attributed to him.</li>
-        </ul></li>
-       <li><b>Stating partial or incomplete provenance</b>.  The newspaper omitted 
-        information about their specific editorial processes in publishing articles.  
-        The chart generator said nothing about the software used, therefore omitting 
-        information about other agents involved in the processes.</li>
-       <li><b>Integrate provenance-related information represented in other vocabularies</b>.  
-        The FOAF vocabulary was used for specifying details about Derek and his company.  
-        The Dublin Core vocabulary was also used. </li>
-       <li><b>Combining distributed provenance records provided by different parties</b>.  
-        The provenance records described in this example came from a government source, 
-        a newspaper, a company, and a blogger.  Using URIs and namespaces, a provenance 
-        record can draw from multiple sources on the Web.</li>
-       <li><b>Describing the commonalities in the derivation of two entities</b>.  
-        The two versions of the charts were both derived through a similar compile activity.</li>
-       <li><b>Relating versions of objects over time</b>.  
-        The second dataset was described as a revision of the first one.</li>
-       <li><b>Stating the provenance of provenance assertions</b>.  
-        Betty could create a provenance record for her blog post, and 
-        then assert the provenance of that provenance record.</li>
-       <li><b>Enabling alternative accounts for the provenance of an entity</b>.  
-        The blog post could have a provenance account from Betty as its author and another 
-        one from the newspaper that the blog quoted from.</li>
-       <li><b>Supporting queries over provenance of different granularity</b>.   
-        The activity of composing the data from different regions was described 
-        in greater detail than the other steps to produce the chart.  
-        The provenance of the chart has much finer-grained detail than 
-        the provenance of the government dataset.</li>
-       <li><b>Supporting reasoning and inference over provenance records</b>.   
-        The provenance assertions explicitly stated above can be used to infer 
-        additional assertions that are true according to the PROV model.  
-        For example, because Derek created the chart acting on behalf of Chart 
-        Generators, a PROV inference may be that the chart can be attributed to Chart Generators.</li>
-      </ol>
-      <p>The examples also illustrate how PROV enables many uses of provenance, 
-       such as checking for possible errors, tracking down parties responsible for 
-       fixing them, repeating processes to re-generate datasets, relating versions of 
-       documents over time, and inspecting sources and attribution.
-      </p>
-     </section>
-     </section>
+   </p>
+  </section>
 
-     <section class="appendix">
-      <h2>Acknowledgements</h2>
-      <p> 
-       This  document has been produced by the Provenance Working Group, and its contents reflect extensive discussion within the Working Group as a whole. The editors extend special thanks to  Sandro Hawke (W3C/MIT) and Ivan Herman (W3C/ERCIM), W3C contacts for the Provenance Working Group.
-      </p> 
-      <p>
-       The editors acknowledge valuable contributions from the following:
-       Tom Baker,
-       David Booth,
-       Bob DuCharme,
-       Robert Freimuth,
-       Satrajit Ghosh,
-       Ralph Hodgson,
-       Renato Iannella,
-       Jacek Kopecky,
-       James Leigh,
-       Chuck Morris,
-       Jacco van Ossenbruggen,
-       Alan Ruttenberg,
-       Reza Samavi, and
-       Antoine Zimmermann.
-      </p>
-      <p>
-       Members of the Provenance Working Group at the time of publication of this document were:
+  <section>
+   <h3>The Complete Example</h3>
+   <p>
+    The full set of examples shown above are available in <a href="primer-turtle-examples.ttl">PROV-O Turtle format</a>
+    or <a href="primer-provn-examples.provn">PROV-N format</a> or <a href="primer-prov-xml-examples.xml">PROV-XML format</a>.
+    We visualize the whole example below.
+   </p>
+   <img src="images/everything.png" alt="Provenance graph for whole example"/>
+  </section>
+ </section>
+ <section>
+  <h3>Summary</h3>
+  <p>The examples in the previous section highlight the following important capabilities of PROV:
+  </p>
+  <ol>
+   <li><b>Representing diverse entities involved in provenance</b>.  
+    The examples illustrate that provenance records can contain different 
+    kinds of information:
+    <ul>
+     <li>The newspaper took an object-centered view on their provenance statements, 
+      stating what objects were derived from others but not stating what processes 
+      were used or what entities were involved.</li>
+     <li>Betty the blogger took a very agent-centered view on provenance.  
+      She chose to associate her blog post with herself, the government agency 
+      that published the data, and the newspaper that published the article.</li>
+     <li>Derek took a process-centered perspective and combined it with object-centered 
+      and agent-centered provenance information. The processes followed to derive the 
+      charts from the government datasets were clearly.  All intermediate and final 
+      documents were stated.  His involvement in the various activities was clearly 
+      recorded, and the resulting documents were attributed to him.</li>
+    </ul></li>
+   <li><b>Stating partial or incomplete provenance</b>.  The newspaper omitted 
+    information about their specific editorial processes in publishing articles.  
+    The chart generator said nothing about the software used, therefore omitting 
+    information about other agents involved in the processes.</li>
+   <li><b>Integrate provenance-related information represented in other vocabularies</b>.  
+    The FOAF vocabulary was used for specifying details about Derek and his company.  
+    The Dublin Core vocabulary was used for specifying the article title. </li>
+   <li><b>Combining distributed provenance records provided by different parties</b>.  
+    The provenance records described in this example came from a government source, 
+    a newspaper, a company, and a blogger.  Using URIs and namespaces, a provenance 
+    record can draw from multiple sources on the Web.</li>
+   <li><b>Describing the commonalities in the derivation of two entities</b>.  
+    The two versions of the charts were both derived through a similar compile activity.</li>
+   <li><b>Relating versions of objects over time</b>.  
+    The second dataset was described as a revision of the first one.</li>
+   <li><b>Stating the provenance of provenance assertions</b>.  
+    Betty could create a provenance record for her blog post, and 
+    then assert the provenance of that provenance record.</li>
+   <li><b>Enabling alternative accounts for the provenance of an entity</b>.  
+    The blog post could have a provenance account from Betty as its author and another 
+    one from the newspaper that the blog quoted from.</li>
+   <li><b>Supporting queries over provenance of different granularity</b>.   
+    The activity of composing the data from different regions was described 
+    in greater detail than the other steps to produce the chart.  
+    The provenance of the chart has much finer-grained detail than 
+    the provenance of the government dataset.</li>
+   <li><b>Supporting reasoning and inference over provenance records</b>.   
+    The provenance assertions explicitly stated above can be used to infer 
+    additional assertions that are true according to the PROV model.  
+    For example, because Derek created the chart acting on behalf of Chart 
+    Generators, a PROV inference may be that the chart can be attributed to Chart Generators.</li>
+  </ol>
+  <p>The examples also illustrate how PROV enables many uses of provenance, 
+   such as checking for possible errors, tracking down parties responsible for 
+   fixing them, repeating processes to re-generate datasets, relating versions of 
+   documents over time, and inspecting sources and attribution.
+  </p>
+ </section>
+</section>
 
-       Ilkay Altintas (Invited expert),
-       Reza B'Far (Oracle Corporation),
-       Khalid Belhajjame (University of Manchester),
-       James Cheney (University of Edinburgh, School of Informatics),
-       Sam Coppens (iMinds - Ghent University),
-       David Corsar (University of Aberdeen, Computing Science),
-       Stephen Cresswell (The National Archives),
-       Tom De Nies (iMinds - Ghent University),
-       Helena Deus (DERI Galway at the National University of Ireland, Galway, Ireland),
-       Simon Dobson (Invited expert),
-       Martin Doerr (Foundation for Research and Technology - Hellas(FORTH)),
-       Kai Eckert (Invited expert),
-       Jean-Pierre EVAIN (European Broadcasting Union, EBU-UER),
-       James Frew (Invited expert),
-       Irini Fundulaki (Foundation for Research and Technology - Hellas(FORTH)),
-       Daniel Garijo (Universidad Politécnica de Madrid),
-       Yolanda Gil (Invited expert),
-       Ryan Golden (Oracle Corporation),
-       Paul Groth (Vrije Universiteit),
-       Olaf Hartig (Invited expert),
-       David Hau (National Cancer Institute, NCI),
-       Sandro Hawke (W3C/MIT),
-       Jörn Hees (German Research Center for Artificial Intelligence (DFKI) Gmbh),
-       Ivan Herman, (W3C/ERCIM),
-       Ralph Hodgson (TopQuadrant),
-       Hook Hua (Invited expert),
-       Trung Dong Huynh (University of Southampton),
-       Graham Klyne (University of Oxford),
-       Michael Lang (Revelytix, Inc.),
-       Timothy Lebo (Rensselaer Polytechnic Institute),
-       James McCusker (Rensselaer Polytechnic Institute),
-       Deborah McGuinness (Rensselaer Polytechnic Institute),
-       Simon Miles (Invited expert),
-       Paolo Missier (School of Computing Science, Newcastle university),
-       Luc Moreau (University of Southampton),
-       James Myers (Rensselaer Polytechnic Institute),
-       Vinh Nguyen (Wright State University),
-       Edoardo Pignotti (University of Aberdeen, Computing Science),
-       Paulo da Silva Pinheiro (Rensselaer Polytechnic Institute),
-       Carl Reed (Open Geospatial Consortium),
-       Adam Retter (Invited Expert),
-       Christine Runnegar (Invited expert),
-       Satya Sahoo (Invited expert),
-       David Schaengold (Revelytix, Inc.),
-       Daniel Schutzer (FSTC, Financial Services Technology Consortium),
-       Yogesh Simmhan (Invited expert),
-       Stian Soiland-Reyes (University of Manchester),
-       Eric Stephan (Pacific Northwest National Laboratory),
-       Linda Stewart (The National Archives),
-       Ed Summers (Library of Congress),
-       Maria Theodoridou (Foundation for Research and Technology - Hellas(FORTH)),
-       Ted Thibodeau (OpenLink Software Inc.),
-       Curt Tilmes (National Aeronautics and Space Administration),
-       Craig Trim (IBM Corporation),
-       Stephan Zednik (Rensselaer Polytechnic Institute),
-       Jun Zhao (University of Oxford),
-       Yuting Zhao (University of Aberdeen, Computing Science).
-     </section>
+<section class="appendix">
+ <h2>Acknowledgements</h2>
+ <p> 
+  This  document has been produced by the Provenance Working Group, and its contents reflect extensive discussion within the Working Group as a whole. The editors extend special thanks to  Sandro Hawke (W3C/MIT) and Ivan Herman (W3C/ERCIM), W3C contacts for the Provenance Working Group.
+ </p> 
+ <p>
+  The editors acknowledge valuable contributions from the following:
+  Tom Baker,
+  David Booth,
+  Bob DuCharme,
+  Robert Freimuth,
+  Satrajit Ghosh,
+  Ralph Hodgson,
+  Renato Iannella,
+  Jacek Kopecky,
+  James Leigh,
+  Chuck Morris,
+  Jacco van Ossenbruggen,
+  Alan Ruttenberg,
+  Reza Samavi, and
+  Antoine Zimmermann.
+ </p>
+ <p>
+  Members of the Provenance Working Group at the time of publication of this document were:
 
-     <section class="appendix"> 
-      <h2>Changes Since Fourth Public Working Draft</h2> 
-      <ul>
-       <li>Corrected XML example for activities</li>
-       <li>Clarified usage of adoptedOnBehalfOf relation</li>
-       <li>Clarified wasQuotedFrom example</li>
-       <li>Adapted PROV-XML examples to clarify they are not RDF/XML</li>
-       <li>Clarified varying sources of example provenance data with different namespace prefixes,
-       and in division of complete example provenance graph</li>
-       <li>Used instance-specific identifiers for entities, activities and agents</li>
-       <li>Added section summarising some additional aspects of PROV not covered in depth</li>
-       <li>Added summary section</li>
-       <li>Added proper acknowledgements</li>
-       <li>Clarified the introduction regarding what is in each section.</li>
-       <li>Typos fixed and minor clarifications throughout</li>
-      </ul>
-     </section>
+  Ilkay Altintas (Invited expert),
+  Reza B'Far (Oracle Corporation),
+  Khalid Belhajjame (University of Manchester),
+  James Cheney (University of Edinburgh, School of Informatics),
+  Sam Coppens (iMinds - Ghent University),
+  David Corsar (University of Aberdeen, Computing Science),
+  Stephen Cresswell (The National Archives),
+  Tom De Nies (iMinds - Ghent University),
+  Helena Deus (DERI Galway at the National University of Ireland, Galway, Ireland),
+  Simon Dobson (Invited expert),
+  Martin Doerr (Foundation for Research and Technology - Hellas(FORTH)),
+  Kai Eckert (Invited expert),
+  Jean-Pierre EVAIN (European Broadcasting Union, EBU-UER),
+  James Frew (Invited expert),
+  Irini Fundulaki (Foundation for Research and Technology - Hellas(FORTH)),
+  Daniel Garijo (Universidad Politécnica de Madrid),
+  Yolanda Gil (Invited expert),
+  Ryan Golden (Oracle Corporation),
+  Paul Groth (Vrije Universiteit),
+  Olaf Hartig (Invited expert),
+  David Hau (National Cancer Institute, NCI),
+  Sandro Hawke (W3C/MIT),
+  Jörn Hees (German Research Center for Artificial Intelligence (DFKI) Gmbh),
+  Ivan Herman, (W3C/ERCIM),
+  Ralph Hodgson (TopQuadrant),
+  Hook Hua (Invited expert),
+  Trung Dong Huynh (University of Southampton),
+  Graham Klyne (University of Oxford),
+  Michael Lang (Revelytix, Inc.),
+  Timothy Lebo (Rensselaer Polytechnic Institute),
+  James McCusker (Rensselaer Polytechnic Institute),
+  Deborah McGuinness (Rensselaer Polytechnic Institute),
+  Simon Miles (Invited expert),
+  Paolo Missier (School of Computing Science, Newcastle university),
+  Luc Moreau (University of Southampton),
+  James Myers (Rensselaer Polytechnic Institute),
+  Vinh Nguyen (Wright State University),
+  Edoardo Pignotti (University of Aberdeen, Computing Science),
+  Paulo da Silva Pinheiro (Rensselaer Polytechnic Institute),
+  Carl Reed (Open Geospatial Consortium),
+  Adam Retter (Invited Expert),
+  Christine Runnegar (Invited expert),
+  Satya Sahoo (Invited expert),
+  David Schaengold (Revelytix, Inc.),
+  Daniel Schutzer (FSTC, Financial Services Technology Consortium),
+  Yogesh Simmhan (Invited expert),
+  Stian Soiland-Reyes (University of Manchester),
+  Eric Stephan (Pacific Northwest National Laboratory),
+  Linda Stewart (The National Archives),
+  Ed Summers (Library of Congress),
+  Maria Theodoridou (Foundation for Research and Technology - Hellas(FORTH)),
+  Ted Thibodeau (OpenLink Software Inc.),
+  Curt Tilmes (National Aeronautics and Space Administration),
+  Craig Trim (IBM Corporation),
+  Stephan Zednik (Rensselaer Polytechnic Institute),
+  Jun Zhao (University of Oxford),
+  Yuting Zhao (University of Aberdeen, Computing Science).
+</section>
 
-     </body></html>
+<section class="appendix"> 
+ <h2>Changes Since Fourth Public Working Draft</h2> 
+ <ul>
+  <li>Corrected XML example for activities</li>
+  <li>Clarified usage of adoptedOnBehalfOf relation</li>
+  <li>Clarified wasQuotedFrom example</li>
+  <li>Adapted PROV-XML examples to clarify they are not RDF/XML</li>
+  <li>Clarified varying sources of example provenance data with different namespace prefixes,
+   and in division of complete example provenance graph</li>
+  <li>Used instance-specific identifiers for entities, activities and agents</li>
+  <li>Added section summarising some additional aspects of PROV not covered in depth</li>
+  <li>Added summary section</li>
+  <li>Added proper acknowledgements</li>
+  <li>Clarified the introduction regarding what is in each section.</li>
+  <li>Typos fixed and minor clarifications throughout</li>
+ </ul>
+</section>
+
+</body></html>
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    style="fill:none;stroke:#000000;stroke-width:0.74672717px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;marker-end:url(#TriangleOutL-1-4-23)"
    d="m 521.39213,800.13911 c -63.57796,40.2577 -63.57796,40.2577 -63.57796,40.2577"
@@ -1959,6 +1997,7 @@
 
 
 
+
 <path
    style="fill:none;stroke:#000000;stroke-width:0.88370097px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;marker-end:url(#TriangleOutL-1-4-23)"
    d="M 547.17205,799.56138 C 501.01243,866.28095 424.25256,884.75204 345.67837,900.36243"
@@ -1983,6 +2022,7 @@
 
 
 
+
 <g
    transform="matrix(1.0693867,0,0,-0.74839875,-193.69309,971.61749)"
    id="g558-3"
@@ -2018,6 +2058,7 @@
 
 
 
+
 </g><path
    style="fill:none;stroke:#000000;stroke-width:0.80000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:2.4, 2.4;stroke-dashoffset:0;marker-start:none"
    d="m -68.624357,802.75059 c -9.693152,-14.21663 -9.693152,-14.21663 -9.693152,-14.21663"
@@ -2057,6 +2098,7 @@
 
 
 
+
 </g><path
    style="fill:none;stroke:#000000;stroke-width:0.80000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:2.4, 2.4;stroke-dashoffset:0;marker-start:none"
    d="m -81.032317,769.3348 c -1.032484,-17.17566 -1.032484,-17.17566 -1.032484,-17.17566"
@@ -2096,6 +2138,7 @@
 
 
 
+
 </g><path
    style="fill:none;stroke:#000000;stroke-width:0.62331414;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:1.8699425, 1.8699425;stroke-dashoffset:0;marker-start:none"
    d="M 48.855922,788.07487 C 47.823432,777.6482 47.823432,777.6482 47.823432,777.6482"
@@ -2128,6 +2171,7 @@
 
 
 
+
 <g
    transform="matrix(0.90807061,0,0,-0.79395312,28.549442,921.59482)"
    id="g558-3-9"
@@ -2163,6 +2207,7 @@
 
 
 
+
 </g><path
    style="fill:none;stroke:#000000;stroke-width:0.79679942px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;marker-end:url(#marker6602)"
    d="M -33.529791,780.89283 C -47.903195,721.89672 -47.903195,721.89672 -47.903195,721.89672"
@@ -2186,6 +2231,7 @@
 
 
 
+
 <path
    style="fill:none;stroke:#000000;stroke-width:0.80704731;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:2.42114196, 2.42114196;stroke-dashoffset:0"
    d="m -44.492325,729.06313 c 97.33151,-0.6462 97.33151,-0.6462 97.33151,-0.6462"
@@ -2232,6 +2278,7 @@
 
 
 
+
 <g
    transform="matrix(0.30786802,0.20888028,0.79449233,-1.2284121,-283.62403,954.67033)"
    id="g572-5-1-7"
@@ -2273,6 +2320,7 @@
 
 
 
+
 <g
    transform="matrix(0.15898884,-0.00230963,0.00875826,-1.4614635,-16.181492,1028.8182)"
    id="g572-5-1-7-0"
@@ -2310,6 +2358,7 @@
 
 
 
+
 <path
    style="fill:none;stroke:#000000;stroke-width:0.6293723px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;marker-end:url(#marker6602)"
    d="m -28.728462,700.07785 c 0,-14.39826 0,-14.39826 0,-14.39826"
@@ -2333,6 +2382,7 @@
 
 
 
+
 <path
    style="fill:none;stroke:#000000;stroke-width:0.79855895px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;marker-end:url(#marker6602-9)"
    d="M 213.76953,780.66181 C 218.16038,704.53278 107.16533,712.22493 3.062825,709.06292"
@@ -2357,6 +2407,7 @@
 
 
 
+
 <path
    style="fill:none;stroke:#000000;stroke-width:0.79445344px;stroke-linecap:butt;stroke-linejoin:miter;stroke-opacity:1;marker-end:url(#marker6602-9)"
    d="M 348.82885,780.01634 C 253.43862,679.42628 121.71857,715.47473 3.0766681,702.33555"
@@ -2381,6 +2432,7 @@
 
 
 
+
 <text
    xml:space="preserve"
    style="font-size:40px;font-style:normal;font-weight:normal;line-height:125%;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;font-family:Sans"
@@ -2398,6 +2450,7 @@
 
 
 
+
 <text
    xml:space="preserve"
    style="font-size:8px;font-style:normal;font-weight:normal;line-height:125%;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;font-family:Sans"
@@ -2416,6 +2469,7 @@
 
 
 
+
 <text
    xml:space="preserve"
    style="font-size:40px;font-style:normal;font-weight:normal;line-height:125%;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;font-family:Sans"
@@ -2433,6 +2487,7 @@
 
 
 
+
 <rect
    style="fill:none;stroke:none"
    id="rect3414"
@@ -2482,6 +2537,7 @@
      id="tspan3495-5">prov:endedAtTime=2012-04-01T15:21:00</tspan></text>
 
 
+
 </g><path
    style="fill:none;stroke:#000000;stroke-width:0.80000001;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:4;stroke-opacity:1;stroke-dasharray:3.2, 0.8;stroke-dashoffset:0"
    d="m -167.37351,1030.6776 c 730.86366,3.231 730.21745,3.231 730.21745,3.231"
@@ -2489,15 +2545,16 @@
    inkscape:connector-curvature="0" /><text
    xml:space="preserve"
    style="font-size:11.19999981px;font-style:normal;font-weight:normal;line-height:125%;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;font-family:Sans"
-   x="430.37085"
+   x="417.5708"
    y="-1053.4406"
    id="text5959"
    sodipodi:linespacing="125%"
    transform="scale(1,-1)"><tspan
      sodipodi:role="line"
-     x="430.37085"
+     x="417.5708"
      y="-1053.4406"
-     id="tspan5965">Betty's provenance data</tspan></text>
+     id="tspan5965">Blogger's provenance data</tspan></text>
+
 
 
 <path
@@ -2518,6 +2575,7 @@
      id="tspan6031">Newspaper's provenance data</tspan></text>
 
 
+
 <text
    xml:space="preserve"
    style="font-size:11.19999981px;font-style:normal;font-weight:normal;text-align:end;line-height:125%;letter-spacing:0px;word-spacing:0px;text-anchor:end;fill:#000000;fill-opacity:1;stroke:none;font-family:Sans"
@@ -2536,6 +2594,7 @@
      id="tspan6035">provenance data</tspan></text>
 
 
+
 <text
    xml:space="preserve"
    style="font-size:11.19999981px;font-style:normal;font-weight:normal;line-height:125%;letter-spacing:0px;word-spacing:0px;fill:#000000;fill-opacity:1;stroke:none;font-family:Sans"
@@ -2550,4 +2609,5 @@
      y="-730.33557">Chart Generators' provenance data</tspan></text>
 
 
+
 </g></svg>
\ No newline at end of file
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/primer/primer-provn-examples.provn	Fri Apr 05 18:13:06 2013 +0100
@@ -0,0 +1,84 @@
+prefix prov <http://www.w3.org/ns/prov#>
+prefix ex <http://www.example.org#>
+prefix dcterms <http://purl.org/dc/terms/>
+prefix foaf <http://xmlns.com/foaf/0.1/>
+
+// The examples below are taken from, and follow the same order
+// as in, the W3C PROV Primer, http://www.w3.org/TR/prov-primer/
+
+// Entities
+
+entity(exn:article, [dcterms:title="Crime rises in cities"])
+entity(exg:dataset1)
+entity(exc:regionList)
+entity(exc:composition1)
+entity(exc:chart1)
+    
+// Activities
+
+activity(exc:compile1)
+activity(exc:compose1)
+activity(exc:illustrate1)
+    
+// Usage and Generation
+
+used(exc:compose1, exg:dataset1, -)
+used(exc:compose1, exc:regionList, -)
+wasGeneratedBy(exc:composition1, exc:compose1, -)
+used(exc:illustrate1, exc:composition1, -)
+wasGeneratedBy(exc:chart1, exc:illustrate1, -)
+    
+// Agents and Responsibility
+
+wasAssociatedWith(exc:compose1, exc:derek, -)
+wasAssociatedWith(exc:illustrate1, exc:derek, -)
+agent(exc:derek,
+      [prov:type='prov:Person', foaf:givenName="Derek", 
+       foaf:mbox="<mailto:[email protected]>"])
+
+agent(exc:chartgen,
+      [prov:type='prov:Organization',
+       foaf:name="Chart Generators Inc"])
+actedOnBehalfOf(exc:derek, exc:chartgen)
+
+wasAttributedTo(exc:chart1, exc:derek)
+    
+// Roles
+
+used(exc:compose1, exg:dataset1, -, [prov:role='exc:dataToCompose'])
+used(exc:compose1, exc:regionList, -, [prov:role='exc:regionsToAggregateBy'])
+wasAssociatedWith(exc:compose1, exc:derek, -, [prov:role='exc:analyst'])
+wasGeneratedBy(exc:composition1, exc:compose1, -, [prov:role='exc:composedData'])
+
+// Derivation and Revision
+
+entity(exg:dataSet2)
+wasDerivedFrom(exg:dataSet2, exg:dataset1, [prov:type='prov:Revision'])
+wasDerivedFrom(exc:chart2, exg:dataSet2)
+entity(exc:chart2)
+wasDerivedFrom(exc:chart2, exc:chart1, [prov:type='prov:Revision'])
+    
+// Plans
+
+activity(exg:correct1)
+agent(exg:edith, [prov:type='prov:Person'])
+entity(exg:instructions)
+wasAssociatedWith(exg:correct1, exg:edith, exg:instructions)
+wasGeneratedBy(exg:dataSet2, exg:correct1, -)
+    
+// Time
+
+wasGeneratedBy(exc:chart1, exc:compile1,  2012-03-02T10:30:00)
+wasGeneratedBy(exc:chart2, exc:compile2, 2012-04-01T15:21:00)
+activity(exg:correct1, 2012-03-31T09:21:00, 2012-04-01T15:21:00)
+    
+// Alternate Entities and Specialization
+
+entity(ex:quoteInBlogEntry-20130326)
+wasDerivedFrom(ex:quoteInBlogEntry-20130326, exn:article, [prov:type='prov:Quotation'])
+    
+entity(exn:articleV1)
+specializationOf(exn:articleV1, exn:article)    
+specializationOf(exn:articleV2, exn:article)
+alternateOf(exn:articleV2, exn:articleV1)
+    
\ No newline at end of file
--- a/primer/primer-provn-examples.txt	Fri Apr 05 18:11:45 2013 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,84 +0,0 @@
-prefix prov <http://www.w3.org/ns/prov#>
-prefix ex <http://www.example.org#>
-prefix dcterms <http://purl.org/dc/terms/>
-prefix foaf <http://xmlns.com/foaf/0.1/>
-
-// The examples below are taken from, and follow the same order
-// as in, the W3C PROV Primer, http://www.w3.org/TR/prov-primer/
-
-// Entities
-
-entity(exn:article, [dcterms:title="Crime rises in cities"])
-entity(exg:dataset1)
-entity(exc:regionList)
-entity(exc:composition1)
-entity(exc:chart1)
-    
-// Activities
-
-activity(exc:compile1)
-activity(exc:compose1)
-activity(exc:illustrate1)
-    
-// Usage and Generation
-
-used(exc:compose1, exg:dataset1, -)
-used(exc:compose1, exc:regionList, -)
-wasGeneratedBy(exc:composition1, exc:compose1, -)
-used(exc:illustrate1, exc:composition1, -)
-wasGeneratedBy(exc:chart1, exc:illustrate1, -)
-    
-// Agents and Responsibility
-
-wasAssociatedWith(exc:compose1, exc:derek, -)
-wasAssociatedWith(exc:illustrate1, exc:derek, -)
-agent(exc:derek,
-      [prov:type='prov:Person', foaf:givenName="Derek", 
-       foaf:mbox="<mailto:[email protected]>"])
-
-agent(exc:chartgen,
-      [prov:type='prov:Organization',
-       foaf:name="Chart Generators Inc"])
-actedOnBehalfOf(exc:derek, exc:chartgen)
-
-wasAttributedTo(exc:chart1, exc:derek)
-    
-// Roles
-
-used(exc:compose1, exg:dataset1, -, [prov:role='exc:dataToCompose'])
-used(exc:compose1, exc:regionList, -, [prov:role='exc:regionsToAggregateBy'])
-wasAssociatedWith(exc:compose1, exc:derek, -, [prov:role='exc:analyst'])
-wasGeneratedBy(exc:composition1, exc:compose1, -, [prov:role='exc:composedData'])
-
-// Derivation and Revision
-
-entity(exg:dataSet2)
-wasDerivedFrom(exg:dataSet2, exg:dataset1, [prov:type='prov:Revision'])
-wasDerivedFrom(exc:chart2, exg:dataSet2)
-entity(exc:chart2)
-wasDerivedFrom(exc:chart2, exc:chart1, [prov:type='prov:Revision'])
-    
-// Plans
-
-activity(exg:correct1)
-agent(exg:edith, [prov:type='prov:Person'])
-entity(exg:instructions)
-wasAssociatedWith(exg:correct1, exg:edith, exg:instructions)
-wasGeneratedBy(exg:dataSet2, exg:correct1, -)
-    
-// Time
-
-wasGeneratedBy(exc:chart1, exc:compile1,  2012-03-02T10:30:00)
-wasGeneratedBy(exc:chart2, exc:compile2, 2012-04-01T15:21:00)
-activity(exg:correct1, 2012-03-31T09:21:00, 2012-04-01T15:21:00)
-    
-// Alternate Entities and Specialization
-
-entity(ex:quoteInBlogEntry-20130326)
-wasDerivedFrom(ex:quoteInBlogEntry-20130326, exn:article, [prov:type='prov:Quotation'])
-    
-entity(exn:articleV1)
-specializationOf(exn:articleV1, exn:article)    
-specializationOf(exn:articleV2, exn:article)
-alternateOf(exn:articleV2, exn:articleV1)
-    
\ No newline at end of file
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/primer/primer-turtle-examples.ttl	Fri Apr 05 18:13:06 2013 +0100
@@ -0,0 +1,116 @@
[email protected] rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#> .
[email protected] prov: <http://www.w3.org/ns/prov#> .
[email protected] exn: <http://www.example.org#> .
[email protected] exg: <http://www.example.org#> .
[email protected] exc: <http://www.example.org#> .
[email protected] exb: <http://www.example.org#> .
[email protected] dcterms: <http://purl.org/dc/terms/> .
[email protected] foaf: <http://xmlns.com/foaf/0.1/> .
+
+# The examples below are taken from, and follow the same order
+# as in the W3C PROV Primer, http://www.w3.org/TR/prov-primer/
+
+# Entities
+
+exn:article      a prov:Entity ;
+                 dcterms:title "Crime rises in cities" .
+exg:dataset1     a prov:Entity .
+exc:regionList   a prov:Entity .
+exc:composition1 a prov:Entity .
+exc:chart1       a prov:Entity .
+
+# Activities
+
+exc:compile1    a prov:Activity .
+exc:compose1    a prov:Activity .
+exc:illustrate1 a prov:Activity .
+
+# Usage and Generation
+
+exc:compose1     prov:used           exg:dataset1 ;
+                 prov:used           exc:regionList .
+exc:composition1 prov:wasGeneratedBy exc:compose1 .
+exc:illustrate1  prov:used           exc:composition1 .
+exc:chart1       prov:wasGeneratedBy exc:illustrate1 .
+
+# Agents and Responsibility
+
+exc:compose1    prov:wasAssociatedWith exc:derek .
+exc:illustrate1 prov:wasAssociatedWith exc:derek .
+exc:derek a prov:Agent ;
+          a prov:Person ;
+          foaf:givenName "Derek"^^xsd:string ;
+          foaf:mbox      <mailto:[email protected]> .
+exc:derek prov:actedOnBehalfOf exc:chartgen .
+exc:chartgen a prov:Agent ;
+             a prov:Organization ;
+             foaf:name "Chart Generators Inc" .
+exc:chart1 prov:wasAttributedTo exc:derek .
+
+# Roles
+    
+exc:dataToCompose        a prov:Role .
+exc:regionsToAggregateBy a prov:Role .
+exc:composedData         a prov:Role .
+exc:analyst              a prov:Role .
+exc:compose1 prov:qualifiedUsage [
+               a prov:Usage ;
+               prov:entity  exg:dataset1 ;
+               prov:hadRole exc:dataToCompose 
+] .    
+exc:compose1 prov:qualifiedUsage [
+               a prov:Usage ;
+               prov:entity  exc:regionList ;
+               prov:hadRole exc:regionsToAggregateBy
+] .
+exc:compose1 prov:qualifiedAssociation [
+               a prov:Association ;
+               prov:agent    exc:derek ;
+               prov:hadRole  exc:analyst
+] .
+exc:composition1 prov:qualifiedGeneration [
+                   a prov:Generation ;
+                   prov:activity  exc:compose1 ;
+                   prov:hadRole   exc:composedData
+] .
+
+# Revision and Derivation
+
+exg:dataSet2 a prov:Entity ;
+             prov:wasRevisionOf exg:dataset1 .
+exc:chart2   a prov:Entity ;
+             prov:wasDerivedFrom exg:dataSet2 .
+exc:chart2   a prov:Entity ;
+             prov:wasRevisionOf exc:chart1 .
+    
+# Plans
+
+exg:correct1    a prov:Activity .
+exg:edith       a prov:Agent, prov:Person .
+exg:instructions a prov:Plan .
+    
+exg:correct1 prov:qualifiedAssociation [
+               a Association ;
+               prov:agent   exg:edith ;
+               prov:hadPlan exg:instructions
+] .
+exg:dataSet2 prov:wasGeneratedBy exg:correct1 .
+    
+# Time
+
+exc:chart1 prov:generatedAtTime "2012-03-02T10:30:00"^^xsd:dateTime .
+exc:chart2 prov:generatedAtTime "2012-04-01T15:21:00"^^xsd:dateTime .
+    
+exg:correct1 prov:startedAtTime "2012-03-31T09:21:00"^^xsd:dateTime ;
+             prov:endedAtTime   "2012-04-01T15:21:00"^^xsd:dateTime .
+    
+# Alternate Entities and Specialization
+
+ex:quoteInBlogEntry-20130326 a prov:Entity ;
+                             prov:wasQuotedFrom exn:article .
+
+exn:articleV1 a prov:Entity ;
+              prov:specializationOf exn:article .
+    
+exn:articleV2 prov:specializationOf exn:article .
+exn:articleV2 prov:alternateOf      exn:articleV1 .
\ No newline at end of file
--- a/primer/primer-turtle-examples.txt	Fri Apr 05 18:11:45 2013 +0100
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,116 +0,0 @@
[email protected] rdf: <http://www.w3.org/1999/02/22-rdf-syntax-ns#> .
[email protected] prov: <http://www.w3.org/ns/prov#> .
[email protected] exn: <http://www.example.org#> .
[email protected] exg: <http://www.example.org#> .
[email protected] exc: <http://www.example.org#> .
[email protected] exb: <http://www.example.org#> .
[email protected] dcterms: <http://purl.org/dc/terms/> .
[email protected] foaf: <http://xmlns.com/foaf/0.1/> .
-
-# The examples below are taken from, and follow the same order
-# as in the W3C PROV Primer, http://www.w3.org/TR/prov-primer/
-
-# Entities
-
-exn:article      a prov:Entity ;
-                 dcterms:title "Crime rises in cities" .
-exg:dataset1     a prov:Entity .
-exc:regionList   a prov:Entity .
-exc:composition1 a prov:Entity .
-exc:chart1       a prov:Entity .
-
-# Activities
-
-exc:compile1    a prov:Activity .
-exc:compose1    a prov:Activity .
-exc:illustrate1 a prov:Activity .
-
-# Usage and Generation
-
-exc:compose1     prov:used           exg:dataset1 ;
-                 prov:used           exc:regionList .
-exc:composition1 prov:wasGeneratedBy exc:compose1 .
-exc:illustrate1  prov:used           exc:composition1 .
-exc:chart1       prov:wasGeneratedBy exc:illustrate1 .
-
-# Agents and Responsibility
-
-exc:compose1    prov:wasAssociatedWith exc:derek .
-exc:illustrate1 prov:wasAssociatedWith exc:derek .
-exc:derek a prov:Agent ;
-          a prov:Person ;
-          foaf:givenName "Derek"^^xsd:string ;
-          foaf:mbox      <mailto:[email protected]> .
-exc:derek prov:actedOnBehalfOf exc:chartgen .
-exc:chartgen a prov:Agent ;
-             a prov:Organization ;
-             foaf:name "Chart Generators Inc" .
-exc:chart1 prov:wasAttributedTo exc:derek .
-
-# Roles
-    
-exc:dataToCompose        a prov:Role .
-exc:regionsToAggregateBy a prov:Role .
-exc:composedData         a prov:Role .
-exc:analyst              a prov:Role .
-exc:compose1 prov:qualifiedUsage [
-               a prov:Usage ;
-               prov:entity  exg:dataset1 ;
-               prov:hadRole exc:dataToCompose 
-] .    
-exc:compose1 prov:qualifiedUsage [
-               a prov:Usage ;
-               prov:entity  exc:regionList ;
-               prov:hadRole exc:regionsToAggregateBy
-] .
-exc:compose1 prov:qualifiedAssociation [
-               a prov:Association ;
-               prov:agent    exc:derek ;
-               prov:hadRole  exc:analyst
-] .
-exc:composition1 prov:qualifiedGeneration [
-                   a prov:Generation ;
-                   prov:activity  exc:compose1 ;
-                   prov:hadRole   exc:composedData
-] .
-
-# Revision and Derivation
-
-exg:dataSet2 a prov:Entity ;
-             prov:wasRevisionOf exg:dataset1 .
-exc:chart2   a prov:Entity ;
-             prov:wasDerivedFrom exg:dataSet2 .
-exc:chart2   a prov:Entity ;
-             prov:wasRevisionOf exc:chart1 .
-    
-# Plans
-
-exg:correct1    a prov:Activity .
-exg:edith       a prov:Agent, prov:Person .
-exg:instructions a prov:Plan .
-    
-exg:correct1 prov:qualifiedAssociation [
-               a Association ;
-               prov:agent   exg:edith ;
-               prov:hadPlan exg:instructions
-] .
-exg:dataSet2 prov:wasGeneratedBy exg:correct1 .
-    
-# Time
-
-exc:chart1 prov:generatedAtTime "2012-03-02T10:30:00"^^xsd:dateTime .
-exc:chart2 prov:generatedAtTime "2012-04-01T15:21:00"^^xsd:dateTime .
-    
-exg:correct1 prov:startedAtTime "2012-03-31T09:21:00"^^xsd:dateTime ;
-             prov:endedAtTime   "2012-04-01T15:21:00"^^xsd:dateTime .
-    
-# Alternate Entities and Specialization
-
-ex:quoteInBlogEntry-20130326 a prov:Entity ;
-                             prov:wasQuotedFrom exn:article .
-
-exn:articleV1 a prov:Entity ;
-              prov:specializationOf exn:article .
-    
-exn:articleV2 prov:specializationOf exn:article .
-exn:articleV2 prov:alternateOf      exn:articleV1 .
\ No newline at end of file
--- a/semantics/prov-sem.html	Fri Apr 05 18:11:45 2013 +0100
+++ b/semantics/prov-sem.html	Fri Apr 05 18:13:06 2013 +0100
@@ -946,7 +946,8 @@
           // only "name" is required. Same format as editors.
  
 //authors:  [] ,
-         
+
+          alternateFormats: [{uri:"prov-sem.pdf",label:"PDF"}],
           // name of the WG
           wg:           "Provenance Working Group",
           
@@ -987,8 +988,9 @@
 data model (called the <dfn>naive semantics</dfn>), viewing
 PROV-DM statements as atomic formulas in the sense of first-order
 logic, and viewing the constraints and inferences specified in
-PROV-CONSTRAINTS as a first-order theory. It is shown that the
-first-order theory is sound with respect to the naive semantics.
+PROV-CONSTRAINTS as a first-order theory. It is shown that valid PROv
+instances (in the sense of PROV-CONSTRAINTS) correspond to satisfiable
+theories.
 This information may be useful to researchers or users of PROV to
 understand the intended meaning and use of PROV for modeling
 information about the actual history, derivation or evolution of Web
@@ -1006,24 +1008,24 @@
 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 [[PROV-OVERVIEW]] 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 [[PROV-OVERVIEW]];</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 [[PROV-PRIMER]];</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 [[PROV-O]];</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 [[PROV-DM]];</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 [[PROV-N]];</li>
+<li> <a href="http://www.w3.org/TR/2013/NOTE-prov-overview-20130430/">PROV-OVERVIEW</a> (Note), an overview of the PROV family of documents [[PROV-OVERVIEW]];</li>
+<li> <a href="http://www.w3.org/TR/2013/NOTE-prov-primer-20130430/">PROV-PRIMER</a> (Note), a primer for the PROV data model [[PROV-PRIMER]];</li>
+<li> <a href="http://www.w3.org/TR/2013/REC-prov-o-20130430/">PROV-O</a> (Recommendation), the PROV ontology, an OWL2 ontology allowing the mapping of PROV to RDF [[PROV-O]];</li>
+<li> <a href="http://www.w3.org/TR/2013/REC-prov-dm-20130430/">PROV-DM</a> (Recommendation), the PROV data model for provenance [[PROV-DM]];</li>
+<li> <a href="http://www.w3.org/TR/2013/REC-prov-n-20130430/">PROV-N</a> (Recommendation), a notation for provenance aimed at human consumption [[PROV-N]];</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
+href="http://www.w3.org/TR/2013/REC-prov-constraints-20130430/">PROV-CONSTRAINTS</a>
+(Recommendation), a set of constraints applying to the PROV
 data model [[PROV-CONSTRAINTS]];</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 [[PROV-XML]];</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 [[PROV-AQ]]; </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 [[PROV-DICTIONARY]];</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 [[PROV-DC]];</li>
+<li> <a href="http://www.w3.org/TR/2013/NOTE-prov-xml-20130430/">PROV-XML</a> (Note),  an XML schema for the PROV data model [[PROV-XML]];</li>
+<li> <a href="http://www.w3.org/TR/2013/NOTE-prov-aq-20130430/">PROV-AQ</a> (Note), the mechanisms for accessing and querying provenance [[PROV-AQ]]; </li>
+<li> <a href="http://www.w3.org/TR/2013/NOTE-prov-dictionary-20130430/">PROV-DICTIONARY</a> (Note) introduces a specific type of collection, consisting of key-entity pairs [[PROV-DICTIONARY]];</li>
+<li> <a href="http://www.w3.org/TR/2013/NOTE-prov-dc-20130430/">PROV-DC</a> (Note) provides a mapping between PROV and Dublic Core Terms [[PROV-DC]];</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
+href="http://www.w3.org/TR/2013/NOTE-prov-sem-20130430/">PROV-SEM</a>
+(Note), a declarative specification in terms of
 first-order logic of the PROV data model (this document);</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 [[PROV-LINKS]].</li>
+<li> <a href="http://www.w3.org/TR/2013/NOTE-prov-links-20130430/">PROV-LINKS</a> (Note) introduces a mechanism to link across bundles [[PROV-LINKS]].</li>
 </ul>
 
 </section>
@@ -1068,7 +1070,7 @@
 <h3>Purpose of this document</h3>
 
 </section>
-<p>The PROV-DM and PROV-CONSTRAINTS give motivating examples that
+<p>The PROV-DM and PROV-CONSTRAINTS specifications give motivating examples that
 provide an intuition about the meaning of the constructs.  For some
 concepts, such as use, start, end, generation/invalidation, and
 derivation, the meaning is either obvious or situation-dependent.
@@ -1091,13 +1093,10 @@
 intent behind certain features of PROV, to researchers investigating
 richer forms of reasoning over provenance, or to future efforts
 building upon PROV.  It is intended as an exploration of <b>one</b> semantics for PROV, not a definitive specification of the  <b>only</b>
-semantics of PROV.  We intend to provide an intuitive semantics that satisfies all
-of the constraints on valid PROV instances, which ensures that no
-invalid PROV instance has a model.  The current naive semantics, however, is
-not complete in the sense that some valid PROV instances lack models.</p>
-<div class="note">
-  <p>TODO: Revise this to reflect future improvements in the semantics.</p>
-  </div>
+semantics of PROV.  We provide a semantics that satisfies all
+of the constraints on valid PROV instances, and such that valid PROV
+instances correspond to satisfiable theories: every valid instance has
+a model, and vice versa.</p>
 
 <p> Although it is a work in progress, the naive semantics has some appealing
 properties.  Specifically, it provides a declarative counterpart to
@@ -1107,7 +1106,7 @@
 to keep the specification closer to implementations, although other
 implementations are possible and allowed.  In addition to providing a
 naive semantics, this document shows that the operational
-presentation of PROV validity checking is sound with respect to the
+presentation of PROV validity checking is equivalent to the
 declarative presentation adopted here.  This could help justify
 alternative approaches to validity checking.</p>
 
@@ -1146,10 +1145,7 @@
   including soundness and a weak form of completeness: a PROV
   instance is valid if and only if it has a naive model.</li>
 </ul>
-<div class="note">
-<p>TODO: We would like to say something stronger here, such as a
-  completeness result for naive models, but this will take more work.</p>
-  </div>
+
 
 </section>
 <section id="audience">
@@ -1162,7 +1158,7 @@
 [[PROV-N]] 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.
+  presented in <a href="http://www.w3.org/TR/2013/REC-prov-constraints-20130430/#overview">Section 2.5</a> of PROV-CONSTRAINTS.
 </p>
 
   <p>This document may be useful to users of PROV who have a formal
@@ -1236,6 +1232,16 @@
 </p>
 </section>
 
+<section>
+<h3> Attributes and Values </h3>
+
+<p>We assume a set $Attributes$ of attribute labels and a set $Values$
+of possible values of attributes.  To allow for the fact that some
+attributes can have undefined or multiple values, we sometimes use the set
+$P(Value)$, that is, the set of sets of values.
+</p> </section>
+
+
 
 <section>
 <h3> Times </h3>
@@ -1244,7 +1250,7 @@
 $Times \subseteq Val$ and $\leq$ is a linear order.
 </p>
 
-
+<!--
 <div class="remark">
 <p>Restricting attention to linearly-ordered times, and imposing this
   order on events, is a simplifying assumption; it is more restrictive than required to model the
@@ -1260,7 +1266,7 @@
   and maximum time, written $min([t_1,t_2]) = t_1$ and $max([t_1,t_2])
   = t_2$ respectively.
 </p>
-
+-->
   
 </section>
   
@@ -1268,15 +1274,6 @@
 
 
   
-<section>
-<h3> Attributes and Values </h3>
-
-<p>We assume a set $Attributes$ of attribute labels and a set $Values$
-of possible values of attributes.  To allow for the fact that some
-attributes can have undefined or multiple values, we sometimes use the set
-$P(Value)$, that is, the set of sets of values.
-</p> </section>
-
 <section id="formulas">
 <h2>Atomic Formulas</h2>
 
@@ -1350,7 +1347,7 @@
   bundles.  Future editions of PROV may incorporate $mentionOf$ or
   other cross-instance assertions, and if so this semantics should be
   generalized in order to provide a rationale for such an
-  extension and to establish the soundness of constaints associated
+  extension and to establish the soundness of constraints associated
   with $mentionOf$. </p>
 </div>
 </section>
@@ -1400,40 +1397,49 @@
 collection of sets, functions and relations.  The components of a
 structure $W$ are given in the rest of the section in
 <em>components</em>, highlighted in boxes.
-<section>
+
+
 
 <h3> Things </h3> 
 
-<p><em>Things</em>  is a set of things in the situation being modeled.  Each thing has a lifetime during which it exists and attributes whose values can change over time.
+<p><em>Things</em>  is a set of things in the situation being modeled.
+Each thing has an associates set of $Events$ and attributes whose
+values can change over time.  Different kinds of $Events$ are specified further below.
 </p>
 <p>To model this, a structure $W$ includes:
 </p>
 <div class="component" id="things"><ol>
   <li> a set $Things$ of things</li>
-  <li> a function $lifetime : Things \to Intervals$ from things to intervals</li>
-  <li>a function $value : Things \times Attributes \times Times \to P(Values)$
+  <li> a set $Events$ of events</li>
+  <li> a function $events : Things \to P(Events)$ from things to
+  sets of events.</li>
+  <li>a function $value : Things \times Attributes \times Events \to
+  P(Values)$ giving the possible values of each attribute of a
+  $Thing$ at the instant of a given event.
 </li>
+<li>Attributes are only defined during the events of a thing, that
+is, $value(T,a,evt) \neq \emptyset$ implies $evt \in events(T)$.
 </ol>
 </div>
 <p>
 The range of $value$ is the set $P(Values)$, indicating that $value$
-is essentially a multi-valued function that returns a set of values (possibly empty).    When $value(x,a,t) =
-\emptyset$, we say that attribute $a$ is undefined for $x$ at time $t$.</p>
+is essentially a multi-valued function that returns a set of values (possibly empty).    When $value(x,a,evt) =
+\emptyset$, we say that attribute $a$ is undefined for $x$ at event $evt$.</p>
 
 <p>Note that this description does not say what the structure of a
-$Thing$ is, only how it may be described in terms of its lifetime
+$Thing$ is, only how it may be described in terms of its events
 and attribute values.  A thing could 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 $\pi$.  All that matters from
-our point of view is that we know how to map the $Thing$ to its time interval and attribute mapping.
+our point of view is that we know how to map the $Thing$ to its events and attribute mapping.
 </p>
 
 
 <p>The identity of a Thing is not observable through its attributes or
-lifetime, so it is possible for two different $Things$ to be indistinguishable by their
-attribute values and lifetime.  That is, if the set of $Things = \{T_0,T_1\}$ and the attributes are
-specified as $value(T_0,a,t) = value(T_1,a,t)$ for each $t\in
-Times$ and $a \in Attributes$, this does not imply that $T_0 = T_1$.
+events, so it is possible for two different $Things$ to be indistinguishable by their
+attribute values and events.  That is, if the set of $Things = \{T_0,T_1\}$ and the attributes are
+specified as $value(T_0,a,evt) = value(T_1,a,evt)$ for each $evt\in
+Events$ and $a \in Attributes$, this does not imply that $T_0 = T_1$.
 </p>
 
 
@@ -1450,7 +1456,7 @@
   that have discrete, fixed features,  and relationships among these
   objects. Some objects, called $Entities$, are associated with
   $Things$, and their fixed attributes need to match those of the
-  associated $Thing$ during their common lifetime.  Others correspond
+  associated $Thing$ during their common events.  Others correspond
   to agents, activities, or identifiable interactions among them.</p>  
 
 <p>In this section, we detail the different subsets of $Objects$, and
@@ -1459,7 +1465,7 @@
 </p>
 
 <p>
-An <em>Object</em> is described by a time interval and attributes with
+An <em>Object</em> is described by a set of events and attributes with
 fixed values.  Objects encompass entities, activities, agents, and
 interactions (i.e., usage, generation, and other events or influence relations).
 To model this, a structure includes:
@@ -1467,13 +1473,13 @@
 
 <div class="component" id="objects">
   <ol><li> a set $Objects$ 
-</li><li> a function $lifetime : Objects \to Intervals$ from objects to time intervals
-</li><li> a function $value : Objects \times Attributes \to P(Values)$
+</li><li> a function $events : Objects \to P(Events)$ from objects
+  to associated sets of events.
+</li><li> a function $value : Objects \times Attributes \to P(Values)$.
 </li></ol>
 </div>
 
-<p>Intuitively, $lifetime(e)$ is the time interval during which object
-$e$ exists.  The set $value(e,a)$ is the set of values of attribute $a$ during the object's lifetime.
+<p>Intuitively, $events(e)$ is the set of events in which $e$ participated.  The set $value(e,a)$ is the set of values of attribute $a$ during the object's events.
 </p>
 
 <p>As with <em>Things</em>, the range of $value$ is sets of values,
@@ -1499,11 +1505,12 @@
   thing. We assume:</p>
 
 <div class="component" id="entities">
-  <ol><li> a set $Entities \subseteq Objects$ of entities, disjoint from $Activities$ and $Events$ below.
+  <ol><li> a set $Entities \subseteq Objects$ of entities, disjoint from $Activities$ below.
 </li><li> a function $thingOf : Entities \to Things$ that associates
-  each $Entity$ $e$ with a $Thing$, such that for each $t \in
-  lifetime(e)$, and for each attribute $a$ we have $value(e,a)
-  \subseteq value(thingOf(e),a,t)$ and $lifetime(e) \subseteq lifetime(thingOf(e))$.
+  each $Entity$ $e$ with a $Thing$, such that $events(e) \subseteq
+  events(thingOf(e))$ and for each $evt \in
+  events(e)$ and for each attribute $a$ we have $value(e,a)
+  \subseteq value(thingOf(e),a,evt)$.  
 </li>
 <!--<li>a relation $SpecializationOf \subseteq Entities \times Entities$
   that is irreflexive and transitive.  Furthermore, if $(e_1,e_2) \in
@@ -1511,7 +1518,7 @@
 <ol><li>
   $thingOf(e_1) = thingOf(e_2)$
   </li>
-  <li>$lifetime(e_1) \subseteq lifetime(e_2)$</li>
+  <li>$events(e_1) \subseteq events(e_2)$</li>
   <li>For each attribute $attr$ we have $value(e_1,attr) \supseteq
   value(e_2,attr)$.</li>
   </ol></li>
@@ -1522,19 +1529,19 @@
 
 <div class="remark"><p> Although both entities and things can have
   undefined or multiple attribute values, their meaning is slightly
-  different: for a thing, $value(x,a,t) = \emptyset$ means that the
-  attribute $a$ has no value at time $t$, whereas for an entity,
+  different: for a thing, $value(x,a,evt) = \emptyset$ means that the
+  attribute $a$ has no value at event $evt$, whereas for an entity,
   $value(x,a) = \emptyset$ only means that the thing associated to
   entity $x$ need not have a
-  fixed value for $a$ during the lifetime of $x$.  This does not imply
-  that $value(thingOf(e),a,t) = \emptyset$ when $t \in lifetime(e)$.
+  fixed value for $a$ during the events of $x$.  This does not imply
+  that $value(thingOf(e),a,evt) = \emptyset$ when $evt \in events(e)$.
   </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 $value(thingOf(e),a,t)$ is $\{1\}$ at
-  some time in $lifetime(e)$ and $value(thingOf(e),a,t') = \{2\}$ at
-  some other time $t'$.  Then $value(e,a)$ must be $\emptyset$ because
+  be present in the associated thing throughout the events of the
+  entity.  For example, suppose $value(thingOf(e),a,evt)$ is $\{1\}$ at
+  some event $ evt \in events(e)$ and $value(thingOf(e),a,evt') = \{2\}$ at
+  some other event $evt'$.  Then $value(e,a)$ must be $\emptyset$ because
   there is no other set of values that is simultaneously contained in
   both $\{1\}$ and $\{2\}$.  </p> </div>
 
@@ -1544,10 +1551,10 @@
   <p>
   In the above description of how $Entities$ relate to $Things$, we
   require  $value(e,a) \subseteq
-  value(thingOf(e),a,t)$ whenever $t \in lifetime(e)$.  Intuitively, this means that if we are
+  value(thingOf(e),a,evt)$ whenever $evt \in events(e)$.  Intuitively, this means that if we are
   talking about a $Thing$ indirectly by describing an $Entity$, then
   any attributes we ascribe to the $Entity$ must also describe the
-  associated $Thing$ during their common lifetime.  Attributes of both
+  associated $Thing$ during their common events.  Attributes of both
   $Entities$ and $Things$ are multi-valued, so there is no
   inconsistency in saying that an entity has two different values for
   some attribute.  In some
@@ -1580,7 +1587,7 @@
   <em>collections</em>, with the following associated structure:</p>
   <div class="component" id="collections">
     <ul><li>A set $Collections \subseteq Entities$</li>
-    <li>A membership relation $MemberOf\subseteq Entities \times Collections$
+    <li>A membership relation $Contains\subseteq Collections \times Entities$
   indicating when an entity is a member of another (collection)
   entity.</li>
   </ul>
@@ -1628,12 +1635,12 @@
 
 <p>We consider a set $Influences \subseteq Objects$ which has disjoint
   subsets
-  <em>Events</em> connecting entities and activities,
-  <em>Associations</em> between agents and activities,
-    <em>Attributions</em> between entities and agents,
-  <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
+  $Events$ connecting entities and activities,
+  $Associations$ between agents and activities,
+    $Attributions$ between entities and agents,
+  $Communications$ between pairs of activities,
+  $Delegations$ between pairs of agents, and
+  $Derivations$ that describe chains of generation and usage
   steps.  These kinds of influences are discussed further below.  Influences are disjoint from entities, activities and agents.
 </p>
 <div class="component" id="influences">
@@ -1654,7 +1661,7 @@
 <section>
 <h5> Events </h5>
 
-<p>An <em>Event</em> is an influence whose lifetime is a single time
+<p>An $Event$ is an influence whose events is a single time
 instant, and relates an activity to an entity (which could be an
 agent).  Events have types including usage, generation, invalidation, starting and ending.  Events are instantaneous.  We introduce:
 </p>
@@ -1662,8 +1669,7 @@
 <ol><li> A set $Events \subseteq Influences$ of events, partitioned
   into disjoint subsets $Starts, Ends, Generations, Usages,
   Invalidations$.
-</li><li> A function $time : Events \to Times$ giving the time of each
-event, such that $lifetime(evt) = \{time(evt)\}$.
+</li><li> A function $time : Events \to Times$.
 </li>
 <li> A quasi-ordering on events $\preceq \subset Events \times
 Events$.  We write $e \prec e'$ when $e \preceq e'$ and $e'
@@ -1686,7 +1692,8 @@
 
 <h5> Associations </h5>
 
-<p>An <em>Association</em> is an influence relating an agent to an activity.  To model associations, we introduce:
+<p>An $Association$ is an influence relating an agent to an activity
+and optional plan.  To model associations, we introduce:
 </p>
 <div class="component" id="associations">
   <p>A set $Associations \subseteq Influences$ with associated
@@ -1698,7 +1705,7 @@
 
 <h5> Attributions </h5>
 
-<p>An <em>Attribution</em> is an influence relating an entity to an agent.  To model associations, we introduce:
+<p>An $Attribution$ is an influence relating an entity to an agent.  To model attributions, we introduce:
 </p>
 <div class="component" id="attributions">
   <p>A set $Attributions \subseteq Influences$ with associated
@@ -1709,12 +1716,12 @@
 </section>
   <section>
   <h5>Communications</h5>
-  <p>A <em>Communication</em> is an influence indicating exchange of
+  <p>A $Communication$ is an influence indicating exchange of
   information between activities.  To model communications, we introduce:
 </p>
 <div class="component" id="communications">
   <p>A set $Communications \subseteq Influences$ with associated
-  function $communicatedBy : Communications \to Activities \times Activities$.
+  function $communicated : Communications \to Activities \times Activities$.
 </p>
   </div>
   
@@ -1722,7 +1729,7 @@
 </section>
   <section>
   <h5>Delegations</h5>
-<p>A <em>Delegation</em> is an influence relating  two agents.  To
+<p>A $Delegation$ is an influence relating  two agents.  To
   model delegations, we introduce:
 </p>
 <div class="component" id="delegations">
@@ -1736,7 +1743,7 @@
   
   <h5> Derivations </h5>
 
-<p>A <em>Derivation</em> is an influence chaining one or more
+<p>A $Derivation$ is an influence chaining one or more
   generation and use steps.  To model derivations, we introduce an
   auxiliary notion of <em>derivation path</em>.  These paths are of the form </p>
 
@@ -1809,120 +1816,190 @@
   <ol>
     <li id="axiom1">
     If $generated(g) = (e,a_1)$ and $used(u) = (a_2,e)$ then there
-    exists $c \in Communications$ such that $communicatedBy(c) = (a_2,a_1)$.
+    exists $c \in Communications$ such that $communicated(c) = (a_2,a_1)$.
     </li>
     <li id="axiom2">
+    If $e \in Entities$ then there exist $gen,inv,a,a'$ such that
+    $generated(gen) = (e,a)$ and $invalidated(inv) = (e,a')$.
+    </li>
+    <li id="axiom3">
     If $started(start) = (a_2,e,a_1)$ then there exists $gen$ such
     that $generated(gen) = (e,a_1)$.
     </li>
-    <li id="axiom3">
+    <li id="axiom4">
     If $ended(end) = (a_2,e,a_1)$ then there exists $gen$ such
     that $generated(gen) = (e,a_1)$.
     </li>
-    <li id="axiom4">
+    <li id="axiom5">
     If $d \in Derivations$ and $prov:Revision \in
     value(d,prov:type)$ and $derivationPath(deriv) = e_2 \cdot w \cdot
     e_1$ then $thingOf(e_1) = thingOf(e_2)$.
     </li>
-    <li id="axiom5">
+    <li id="axiom6">
     If $attributedTo(att) = (e,ag)$ then there exist $gen$ and $assoc$
     such that $generated(gen) = (e,a)$ and $associatedWith(assoc) = (a,ag)$.
     </li>
-    <li id="axiom6">
+    <li id="axiom7">
     If $actedFor(deleg) = (ag_2,ag_1,act)$ then there exist
-    $assoc_1,assoc_2,pl_1,pl_2$ such that $associatedWith(assoc_1) = (act,ag_1,pl_1)$
-    and $associatedWith(assoc_2) = (act,ag_2,pl_2)$.
-    </li>
-    <li id="axiom7">
-    If $generated(id) = (e,a)$ then $influenced(id) = (e,a)$.
+    $assoc_1,assoc_2,pl_1,pl_2$ such that $associatedWith(assoc_1) = (ag_1,act,pl_1)$
+    and $associatedWith(assoc_2) = (ag_2,act,pl_2)$.
     </li>
     <li id="axiom8">
-        If $used(id) = (e,a)$ then $influenced(id) = (e,a)$.
+    If $generated(id) = (e,a)$ then $influenced(id) = (e,a)$.
     </li>
     <li id="axiom9">
-            If $communicatedBy(id) = (a_2,a_1)$ then $influenced(id) = (a_2,a_1)$.
+        If $used(id) = (e,a)$ then $influenced(id) = (e,a)$.
     </li>
     <li id="axiom10">
-     If $started(id) = (a_2,e,a_1)$ then $influenced(id) = (a_2,e)$.
+            If $communicated(id) = (a_2,a_1)$ then $influenced(id) = (a_2,a_1)$.
     </li>
     <li id="axiom11">
-         If $ended(id) = (a_2,e,a_1)$ then $influenced(id) = (a_2,e)$.
+     If $started(id) = (a_2,e,a_1)$ then $influenced(id) = (a_2,e)$.
     </li>
     <li id="axiom12">
+         If $ended(id) = (a_2,e,a_1)$ then $influenced(id) = (a_2,e)$.
+    </li>
+    <li id="axiom13">
     If $invalidated(id) = (e,a)$ then $influenced(id) = (e,a)$.
     </li>
-    <li id="axiom13">
+    <li id="axiom14">
     If $derivationPath(id) = e_2 \cdot w \cdot e_1$ then
     $influenced(id) = (e_2,e_1)$.
     </li>
-    <li id="axiom14">
+    <li id="axiom15">
     If $attributedTo(id) = (e,ag)$ then $influenced(id) = (e,ag)$.
     </li>
-    <li id="axiom15">
-    If $associatedWith(id) = (a,ag,pl)$ then $influenced(id) = (a,ag)$.
-    </li>
     <li id="axiom16">
-    If $actedFor(id) = (ag_2,ag_1)$ then $influenced(id) = (ag_2,ag_1)$.
+    If $associatedWith(id) = (a,ag,pl)$ then $influenced(id) = (a,ag)$.
     </li>
     <li id="axiom17">
-    If $generate(gen) = (e,a) = generated(gen')$ then $gen = gen'$.
+    If $actedFor(id) = (ag_2,ag_1)$ then $influenced(id) = (ag_2,ag_1)$.
     </li>
     <li id="axiom18">
-    If $invalidated(inv) = (e,a) = invalidated(inv')$ then $inv=inv'$.
+    If $generated(gen) = (e,a) = generated(gen')$ then $gen = gen'$.
     </li>
     <li id="axiom19">
-    If $started(st) = (a,e_1,a')$ and $started(st') = (a,e_2,a')$ then $st=st'$.
+    If $invalidated(inv) = (e,a) = invalidated(inv')$ then $inv=inv'$.
     </li>
     <li id="axiom20">
-    If $ended(end) = (a,e_1,a')$ and $ended(end') = (a,e_2,a')$ then $end=end'$.
+    If $started(st) = (a,e_1,a')$ and $started(st') = (a,e_2,a')$ then $st=st'$.
     </li>
     <li id="axiom21">
-    If $started(st) = (a,e)$ then $st \preceq evt$ for all $evt \in events(a)$.
+    If $ended(end) = (a,e_1,a')$ and $ended(end') = (a,e_2,a')$ then $end=end'$.
     </li>
     <li id="axiom22">
-        If $ended(end) = (a,e,a') $ then $evt \preceq end$ for all $evt \in events(a)$.
+    If $started(st) = (a,e)$ then $st \preceq evt$ for all $evt \in
+    events(a) - Invalidations$.
     </li>
     <li id="axiom23">
-    If $generated(gen) = (e,a)$ then $gen \preceq evt$ for all $evt \in events(e)$.
+        If $ended(end) = (a,e,a') $ then $evt \preceq end$ for all
+    $evt \in events(a) - Invalidations$.
     </li>
     <li id="axiom24">
-        If $invalidated(inv) = (e,a)$ then $evt\preceq inv$ for all $evt \in events(e)$.
+    If $generated(gen) = (e,a)$ then $gen \preceq evt$ for all $evt \in events(e)$.
     </li>
     <li id="axiom25">
+        If $invalidated(inv) = (e,a)$ then $evt\preceq inv$ for all
+    $evt \in events(e)$.
+    </li>
+    <li id="axiom26">
     For any derivation $deriv$, with path $derivationPath(deriv) = w$,
     if $e_2 \cdot g \cdot a \cdot u \cdot e_1 $ is a substring of $w$
     where $e_1,e_2 \in Entities$, $g \in Generations$, $u \in Usages$
     and $a \in Activities$ then $u \preceq g$.
     </li>
-    <li id="axiom26">
+    <li id="axiom27">
     For any derivation $deriv$, with path $derivationPath(deriv) = e_2
     \cdot w \cdot e_1$, if $generated(gen_1) = (e_1,a_1)$ and
     $generated(gen_2) = (e_2,a_2)$ then $gen_1 \prec gen_2$.  
     </li>
+    <li id="axiom28">
+    If  $associatedWith(assoc) = (a,ag,pl)$  and $started(start) = (a,e_1,a_1)$ and $invalidated(inv) =
+    (ag,a_2)$ then $start \preceq inv$.
+    </li>
+    <li id="axiom29">
+    If  $associatedWith(assoc) = (a,ag,pl)$  and $generated(gen) =
+    (ag,a_1)$ and $ended(end) = (a,e_2,a_2)$ then $gen \preceq end$.
+    </li>
+    <li id="axiom30">
+    If  $associatedWith(assoc) = (a,ag,pl)$  and $started(start) = (a,e_1,a_1)$ and $ended(end) =
+    (ag,e_2,a_2)$ then $start \preceq end$.
+    </li>
+    <li id="axiom31">
+    If  $associatedWith(assoc) = (a,ag,pl)$  and $started(start) =
+    (ag,e_1,a_1)$ and $ended(end) = (a,e_2,a_2)$ then $start \preceq end$.
+    </li>
+       <li id="axiom32">
+    If $attributedTo(attrib) = (e,ag)$  and $generated(gen_1) =
+    (ag_1,a_1)$ and $generated(gen_2) = (e,a_2)$ then $gen_1 \preceq gen_2$.
+    </li>
+       <li id="axiom33">
+    If $attributedTo(attrib) = (e,ag)$  and $started(start) =
+    (ag_1,e_1,a_1)$ and $generated(gen) = (e,a_2)$ then $start \preceq gen$.
+    </li>
+       <li id="axiom34">
+    If $actedFor(deleg) = (ag_2,ag_1,a)$  and $generated(gen) =
+    (ag_1,a_1)$ and $invalidated(inv) = (ag_2,a_2)$ then $gen \preceq inv$.
+    </li>
+       <li id="axiom35">
+    If $actedFor(deleg) = (ag_2,ag_1,a)$ and $started(start) =
+    (ag_1,e_1,a_1)$ and $ended(end) = (ag_2,e_2,a_2)$ then $start \preceq
+    end$.
+    </li>
     </ol>
 </div>
 
     <p>These properties are called <em>axioms</em>, and they are
   needed to ensure that the PROV-CONSTRAINTS inferences and
   constraints hold in all structures.</p>
+
+<div class="remark">
+  <p> Axioms 22 and 23 do not require that invalidation events
+  originating from an activity follow the activity's start
+  event(s) or precede its end event(s).
+  This is because
+  there is no such constraint in PROV-CONSTRAINTS.  Arguably, there
+  should be a constraint analogous to Constraint 34 that specifies
+  that any invalidation event in which an activity participates must
+  follow the activity's start event(s) and precede its end event(s).
+  </div>
   </section>
 
 <section>
 <h3> Putting it all together </h3>
 
-<p>A <em>structure</em> $W$ is a collection of sets, functions, and relations containing all of the above
-described components.  If we need to talk about the objects or relations of
+<p>A <em>PROV structure</em> $W$ is a collection of sets, functions, and relations containing all of the above
+described components and satisfying all of the associated properties
+and axions.  If we need to talk about the objects or relations of
 more than one structure then we may write $W_1.Objects$, $W_1.Things$,
 etc.; otherwise, to
 decrease notational clutter, when we consider a fixed structure then the names of the sets, relations and functions above refer to the components of that model.
 </p>
 
 
-
+<p>
+Some features of PROV structures are relatively obvious or routine,
+corresponding directly to features of PROV and associated inferences.
+For example, the functions $used, generated, invalidated, started,
+ended$ mapping events to their associated entities or activities, and
+$communicated, associatedWith, attributedTo, actedFor$ associating
+other types of influences with appropriate data. 
+</p>
+  <p>
+  On the other hand,
+some features are more distinctive, and represent areas where formal
+modeling has been used to guide the development of PROV.  Derivation
+paths are one such distinctive feature; they correspond to an
+intuition that derivations may describe one or multiple generation-use
+steps leading from one entity to another.  Another distinctive feature
+is the use of $Things$, which correspond to changing, real-world
+things, as opposed to $Entities$, which correspond to limited views or
+perspectives on $Things$, with some fixed aspects.  The semantic
+structures of $Things$ and $Entities$ provides a foundation for the
+$alternateOf$ and $specializationOf$ relations.
+</p>
   
-<div class="note">
-  TODO: Highlight the distinctive vs obvious/routine features.
-  </div>
+
 
 </section>
 <section>
@@ -2026,8 +2103,6 @@
 <ol>
 <li>[WF] $id$ denotes an entity $ent = \rho(id) \in Entities$
 </li>
-<li>There exists $gen,a$ such that $generated(gen) = (e,a)$.</li>
-<li>There exists $inv,a'$ such that $invalidated(inv) = (e,a)$.</li>
 <li>the attributes match: $match(W,ent, attrs)$.
 </li>
 </ol>
@@ -2304,7 +2379,7 @@
 <li>[WF] $a_1,a_2$ denote  activities $act_1 = \rho(a_1) \in
 Activities, act_2 = \rho(a_2)\in Activities$.
 </li>
-<li>There exist $gen,use,ent$ such that $communicatedBy(comm) =
+<li>There exist $gen,use,ent$ such that $communicated(comm) =
 (act_2,act_1)$ and $generated(gen) = (ent,act_1)$ and $used(use) = (act_2,ent)$.
 </li>
 <li>The attributes match: $match(W,comm,attrs)$.
@@ -2429,18 +2504,18 @@
   $W,\rho \models specializationOf(e_1,e_2)$ holds if and only if:</p>
 <ol>
 <li>[WF] Both $e_1$ and $e_2$ are entity identifiers, denoting
-  distinct entities $ent_1
+  entities $ent_1
   = \rho(e_1) \in Entities$ and $ent_2 = \rho(e_2) \in Entities$.
 </li>
 <!--<li>$(ent_1,ent_2) \in SpecializationOf$.-->
   <li>The two entities present aspects of the same thing, that is, $thingOf(ent_1) = thingOf(ent_2)$.
 </li>
-<li>The lifetime of $ent_1$ is contained in that of $ent_2$, i.e. $lifetime(ent_1) \subseteq lifetime(ent_2)$.
+<li>The events of $ent_1$ is contained in that of $ent_2$, i.e. $events(ent_1) \subseteq events(ent_2)$.
 </li>
 <li>For each attribute $attr$ we have $value(ent_1,attr) \supseteq value(ent_2,attr)$.</li>
 <li>At least one of these inclusions is strict: that is, either
-  $lifetime(ent_1) \subsetneq lifetime(ent_2)$ or for some $attr$ we
- value(ent_1,attr) \supsetneq value(ent_2,attr)$.
+  $events(ent_1) \subsetneq events(ent_2)$ or for some $attr$ we have
+ $value(ent_1,attr) \supsetneq value(ent_2,attr)$.
 </li>
 </ol>
 </div>
@@ -2451,18 +2526,19 @@
   <li>The two entities are different: $ent_1 \neq ent_2$.
   <li>The two Entities refer to the same Thing, that is, $thingOf(ent_1) = thingOf(ent_2)$.
 </li>
-<li>The lifetime of $ent_1$ is contained in that of $ent_2$, i.e. $lifetime(ent_1) \subseteq lifetime(ent_2)$.
+<li>The events of $ent_1$ is contained in that of $ent_2$, i.e. $events(ent_1) \subseteq events(ent_2)$.
 </li>
 <li>For each attribute $attr$ we have $value(ent_1,attr) \supseteq value(ent_2,attr)$.
 </li></ol>
 -->
 <p>The second criterion says that the two Entities present (possibly different) aspects of
 the same Thing. Note that the third criterion allows $ent_1$ and
-$ent_2$ to have the same lifetime (or that of $ent_2$ can be larger).
+$ent_2$ to have the same events (or that of $ent_2$ can be larger).
 The last criterion allows $ent_1$ to have more defined attributes than
 $ent_2$, but they must include the attributes defined by $ent_2$.  Two
   different entities that have the same attributes can also be related
-  by specialization.  
+  by specialization.  The fifth criterion (indirectly) ensures that
+  specialization is irreflexive.
 </p>
 </div>
 </section>
@@ -2500,7 +2576,7 @@
   = \rho(c) \in Collections$ and $ent = \rho(e) \in Entities$.
 </li>
 <li>The entity $ent$ is a member of the collection $coll$: that is,
-$(ent,coll) \in MemberOf$.
+$(coll,ent) \in Contains$.
 </li></ol>
 </div>
 
@@ -2537,7 +2613,7 @@
   </div>
 
   <div class="remark">
-    <p> Although times are linearly ordered, the time ordering is
+    <p> The ordering of time values associated to events is
     unrelated to the event ordering. For example:</p>
     <pre>
 entity(e)
@@ -2585,7 +2661,7 @@
       $\rho(c) \in Collections$.</li>
       <li>$W,\rho\models typeOf(c,EmptyCollection)$ holds if and only if
       $\rho(c) \in Collections$ and there is no $e \in Entities$ such
-    that $(e,\rho(c)) \in MemberOf$.</li>
+    that $(\rho(c),e) \in  Contains$.</li>
       </ol>
       
     </div>
@@ -2642,9 +2718,8 @@
 \end{array}$</div>
 
 <div class="proof">
-  <p> This follows from the semantics of entity formulas, specifically
-  the requirement that generation and invalidation events exist for
-  the entity.</p>
+  <p> This follows from <a href="#axiom2">Axiom 2</a>, which
+  requires that generation and invalidation events exist for each entity.</p>
   </div>
 <div class="inference" number="8" id="activity-start-end-inference">$\begin{array}[t]{l}
 \forall a,t_1,t_2,attrs.~
@@ -2672,7 +2747,7 @@
 \end{array}$</div>
 
 <div class="proof">
-  <p>This follows from <a href="#axiom2">Axiom 2</a>.</p>
+  <p>This follows from <a href="#axiom3">Axiom 3</a>.</p>
   </div>
 <div class="inference" number="10" id="wasEndedBy-inference">$\begin{array}[t]{l}
 \forall id,a,e_1,a_1,t,attrs.~
@@ -2684,7 +2759,7 @@
 \end{array}$</div>
 
 <div class="proof">
-  <p>This follows from <a href="#axiom3">Axiom 3</a>.</p>
+  <p>This follows from <a href="#axiom4">Axiom 4</a>.</p>
   </div>
 <div class="inference" number="11"
   id="derivation-generation-use-inference">
@@ -2713,7 +2788,7 @@
 \end{array}$</div>
 <div class="proof">
   <p> This follows from the semantics of derivation steps (precise or
-  imprecise) and <a href="#axiom4">Axiom 4</a>.</p>
+  imprecise) and <a href="#axiom5">Axiom 5</a>.</p>
   </div>
 <div class="inference" number="13" id="attribution-inference">$\begin{array}[t]{l}
 \forall att,e,ag,attrs.~
@@ -2725,7 +2800,7 @@
 \end{array}$</div>
 <div class="proof">
   <p>This follows from the semantics of generation, association, and
-  attribution, by <a href="#axiom5">Axiom 5</a></p>
+  attribution, by <a href="#axiom6">Axiom 6</a>.</p>
   </div>
   
 <div class="inference" number="14" id="delegation-inference">$\begin{array}[t]{l}
@@ -2737,7 +2812,7 @@
 \exists id_1,pl_1,id_2,pl_2.~wasAssociatedWith(id_1,a,ag_1,pl_1,[]) \wedge wasAssociatedWith(id_2,a,ag_2,pl_2,[])
 \end{array}$</div>
 <div class="proof">
-  <p>This follows from the semantics of association and delegation, by <a href="#axiom6">Axiom 6</a></p>
+  <p>This follows from the semantics of association and delegation, by <a href="#axiom7">Axiom 7</a>.</p>
   </div>
 <div class="inference" number="15" id="influence-inference"><ol><li>$\begin{array}[t]{l}
 \forall id,e,a,t,attrs.~
@@ -2814,7 +2889,7 @@
 \end{array}$</li></ol></div>
 
 <div class="proof">
-  <p>This follows via <a href="#axiom7">Axioms 7</a> through <a href="#axiom16">16</a>.
+  <p>This follows via <a href="#axiom8">Axioms 8</a> through <a href="#axiom17">17</a>.
   </div>
 <div class="inference" number="16" id="alternate-reflexive">$\begin{array}[t]{l}
 \forall e.~
@@ -2869,8 +2944,8 @@
 <div class="proof">
   <p> Suppose the conditions for specialization hold of $ent_1$ and
   $ent_2$ and for $ent_2$ and $ent_3$, where $ent_1 = \rho(e_1)$ and $ent_2 = \rho(e_2)$ and $ent_3 =
-  \rho(e_3)$. Then $lifetime(e_1) \subseteq lifetime(e_2) \subseteq
-  lifetime(e_3)$.  Moreover, 
+  \rho(e_3)$. Then $events(e_1) \subseteq events(e_2) \subseteq
+  events(e_3)$.  Moreover, 
   $value(obj_2,attr) \supseteq value(obj_3,attr)$, and similarly
   $value(obj_1,attr)\supseteq value(obj_2,attr)$ so $value(obj_1,attr)
   \supseteq value(obj_3,attr)$.  Finally, at least one of the
@@ -2909,10 +2984,9 @@
   $entity(e_1,attrs)$ holds, we know that $v \in value(ent_1,att)$.
   Thus $v \in value(ent_2,att)$ since $value(ent_2,att) \supseteq value(ent_1,att)$.  Since
   this is the case for all attribute-value pairs in $attrs$, and since
-  $e_2$ obviously denotes an entity, we can conclude $W,\rho \models entity(e,attrs$).
+  $e_2$ obviously denotes an entity, we can conclude $W,\rho \models entity(e_2,attrs)$.
   </p>
 </div>
-
 <section>
 <h2>Constraints</h2>
 <section>
@@ -3019,7 +3093,7 @@
 \end{array}$</div>
 <div class="proof">
   <p>
-  This follows from <a href="#axiom17">Axiom 17</a>.
+  This follows from <a href="#axiom18">Axiom 18</a>.
   </p>
   </div>
 <div class="constraint" number="25" id="unique-invalidation">$\begin{array}[t]{l}
@@ -3032,7 +3106,7 @@
 \end{array}$</div>
 <div class="proof">
   <p>
-  This follows from <a href="#axiom17">Axiom 18</a>.
+  This follows from <a href="#axiom19">Axiom 19</a>.
   </p>
   </div>
 <div class="constraint" number="26" id="unique-wasStartedBy">$\begin{array}[t]{l}
@@ -3045,7 +3119,7 @@
 \end{array}$</div>
 <div class="proof">
   <p>
-  This follows from <a href="#axiom17">Axiom 19</a>.
+  This follows from <a href="#axiom20">Axiom 20</a>.
   </p>
   </div>
 
@@ -3059,7 +3133,7 @@
 \end{array}$</div>
 <div class="proof">
   <p>
-  This follows from <a href="#axiom17">Axiom 20</a>.
+  This follows from <a href="#axiom21">Axiom 21</a>.
   </p>
   </div>
 <div class="constraint" number="28" id="unique-startTime">$\begin{array}[t]{l}
@@ -3105,7 +3179,7 @@
 start \precedes end
 \end{array}$</div>
 <div class="proof">
-  <p>This follows from <a href="#axiom21">Axiom 21</a>.
+  <p>This follows from <a href="#axiom22">Axiom 22</a>.
   </p>
   </div>
   
@@ -3118,7 +3192,7 @@
 start_1 \precedes start_2
 \end{array}$</div>
 <div class="proof">
-  <p>This follows from <a href="#axiom21">Axiom 21</a>.
+  <p>This follows from <a href="#axiom22">Axiom 22</a>.
   </p>
   </div>
   
@@ -3151,8 +3225,8 @@
 use \precedes end
 \end{array}$</li></ol></div>
 <div class="proof">
-  <p>Part 1 follows from <a href="#axiom21">Axiom 21</a> and part 2
-  follows from <a href="#axiom22">Axiom 22</a>.
+  <p>Part 1 follows from <a href="#axiom22">Axiom 22</a> and part 2
+  follows from <a href="#axiom23">Axiom 23</a>.
   </p>
   </div>
   
@@ -3172,8 +3246,8 @@
 gen \precedes end
 \end{array}$</li></ol></div>
 <div class="proof">
-  <p>Part 1 follows from <a href="#axiom21">Axiom 21</a> and part 2
-  follows from <a href="#axiom22">Axiom 22</a>.
+  <p>Part 1 follows from <a href="#axiom22">Axiom 22</a> and part 2
+  follows from <a href="#axiom23">Axiom 23</a>.
   </p>
   </div>
   
@@ -3203,7 +3277,7 @@
 \end{array}$</div>
 
 <div class="proof">
-  <p>This follows from <a href="#axiom22">Axiom 22</a>.
+  <p>This follows from <a href="#axiom23">Axiom 23</a>.
   </p>
   </div>
 
@@ -3217,7 +3291,7 @@
 \end{array}$</div>
 
 <div class="proof">
-  <p>This follows from <a href="#axiom23">Axiom 23</a>.
+  <p>This follows from <a href="#axiom24">Axiom 24</a>.
   </p>
   </div>
 
@@ -3231,7 +3305,7 @@
 \end{array}$</div>
 
 <div class="proof">
-  <p>This follows from <a href="#axiom24">Axiom 24</a>.
+  <p>This follows from <a href="#axiom25">Axiom 25</a>.
   </p>
   </div>
 
@@ -3245,7 +3319,7 @@
 \end{array}$</div>
 
 <div class="proof">
-  <p>This follows from <a href="#axiom23">Axiom 23</a>.
+  <p>This follows from <a href="#axiom24">Axiom 24</a>.
   </p>
   </div>
 
@@ -3259,7 +3333,7 @@
 \end{array}$</div>
 
 <div class="proof">
-  <p>This follows from <a href="#axiom24">Axiom 24</a>.
+  <p>This follows from <a href="#axiom25">Axiom 25</a>.
   </p>
   </div>
 
@@ -3274,7 +3348,7 @@
 \end{array}$</div>
 
 <div class="proof">
-  <p>This follows from <a href="#axiom25">Axiom 25</a>.
+  <p>This follows from <a href="#axiom26">Axiom 26</a>.
   </p>
   </div>
 
@@ -3288,7 +3362,7 @@
 \end{array}$</div>
 
 <div class="proof">
-  <p>This follows from <a href="#axiom25">Axiom 26</a>.
+  <p>This follows from <a href="#axiom27">Axiom 27</a>.
   </p>
   </div>
 
@@ -3309,8 +3383,8 @@
 \end{array}$</li></ol></div>
 
 <div class="proof">
-  <p>Part 1 follows from <a href="#axiom23">Axiom 23</a>.  Part 2
-  follows from <a href="#axiom24">Axiom 24</a>.
+  <p>Part 1 follows from <a href="#axiom24">Axiom 24</a>.  Part 2
+  follows from <a href="#axiom25">Axiom 25</a>.
   </p>
   </div>
 
@@ -3331,8 +3405,8 @@
 \end{array}$</li></ol></div>
 
 <div class="proof">
-  <p>Part 1 follows from <a href="#axiom23">Axiom 23</a>.  Part 2
-  follows from <a href="#axiom24">Axiom 24</a>.
+  <p>Part 1 follows from <a href="#axiom24">Axiom 24</a>.  Part 2
+  follows from <a href="#axiom25">Axiom 25</a>.
   </p>
   </div>
 
@@ -3347,7 +3421,10 @@
 
 
 <div class="proof">
-  <p> ???
+  <p> This follows from <a href="#axiom24">Axiom 24</a> and the fact
+  that if $e_2$ specializes $e_1$ then all of the events of the $e_2$
+  are events of $e_1$.  Thus, the generation of $e_1$ precedes all
+  events of $e_2$.
   </p>
   </div>
 
@@ -3361,7 +3438,10 @@
 \end{array}$</div>
 
 <div class="proof">
-  <p> ???
+  <p> This follows from <a href="#axiom25">Axiom 25</a> and the fact
+  that if $e_2$ specializes $e_1$ then all of the events of the $e_2$
+  are events of $e_1$.  Thus, the invalidation of $e_1$ follows all
+  events of $e_2$.
   </p>
   </div>
 
@@ -3396,7 +3476,8 @@
 \end{array}$</li></ol></div>
 
 <div class="proof">
-  <p>TODO - arbitrary axioms.
+  <p>The four parts follow from <a href="#axiom28">Axiom 28</a> through
+  <a href="#axiom31">Axiom 31</a> respectively.
   </p>
   </div>
 
@@ -3417,7 +3498,8 @@
 \end{array}$</li></ol></div>
 
 <div class="proof">
-  <p>TODO - arbitrary axioms.
+  <p>These properties follow from <a href="#axiom32">Axiom 32</a> and
+  <a href="#axiom33">Axiom 33</a>.
   </p>
   </div>
 
@@ -3438,7 +3520,8 @@
 \end{array}$</li></ol></div>
 
 <div class="proof">
-  <p>TODO - arbitrary axioms.
+  <p>These properties follow from <a href="#axiom34">Axiom 34</a> and
+  <a href="#axiom35">Axiom 35</a>.
   </p>
   </div>
 
@@ -3629,8 +3712,8 @@
 \end{array}$</div>
 <div class="proof">
   <p>This follows from the fact that in the semantics of
-  $specializationOf$, the two entities denoted by the first and second
-  arguments are required to be distinct.
+  $specializationOf$, the requirement that one of the inclusions is
+  strict implies that the two entities cannot be the same.
   </p>
   </div>
 <div class="constraint" number="53" id="impossible-property-overlap"><p>For each $r$  and  $s \in \{ used, wasGeneratedBy, wasInvalidatedBy, wasStartedBy, wasEndedBy, wasInformedBy, wasAttributedTo, wasAssociatedWith, actedOnBehalfOf\}$ such that $r$  and  $s$ are different relation names, the following constraint holds:</p>$\begin{array}[t]{l}
@@ -3642,7 +3725,7 @@
 False
 \end{array}$</div>
 <div class="proof">
-  <p>This follows from the assumption that the different classes of
+  <p>This follows from the assumption that the different kinds of
   influences are disjoint sets, characterized by their types.
   </p>
   </div>
@@ -3705,8 +3788,9 @@
 <p>Our main soundness result is:</p>
 
 <div class="theorem" id="soundness-theorem">
-  <p> Let $W$ be a PROV model, that is, a structure satisfying all of
-  the axioms</p>.
+  <p> Let $W$ be a PROV structure, that is, a structure providing all
+  of the components above and satisfying all of
+  the axioms.</p>
   <ol>
   <li>If $I$ is an instance and $W \models I$ and $I'$ is obtained from $I$ by applying one
   of the PROV inferences, then $W \models I'$.</li>
@@ -3724,7 +3808,8 @@
   or uniqueness constraint steps: if $I$ is in normal form then we are
   done. If $I$ is not in normal form then if an inference is  applicable, then use part 1; if a uniqueness constraint is
   applicable, then from $W \models I$ the uniqueness constraint cannot
-  fail on $I$ and $W \models I'$.</p>
+  fail on $I$ and $W \models I'$, as argued for the key constraints in
+  the previous section.</p>
 <p>
   For part 3, the arguments are as
   in the previous section for each constraint. </p>
@@ -3745,28 +3830,40 @@
   <p> Let $I$ be a valid PROV instance that is in normal form.
   We define a structure $M(I)$ as follows, by giving the sets,
   functions and relations specified in the components in <a
-  href="#structures">Section 3</a>.</p>
+  href="#structures">Section 3</a>, and finally verifying that the
+  axioms hold.</p>
+
+  <p>First, without loss of generality, we assume that all times
+  specified in activity or event formulas in $I$ are ground values.
+  If not, set each variable in such a position to some dummy value.</p>
 
   <section>
 <h4>Sets</h4>
   <p> The sets of structure $M(I)$ are: </p>
-
-  \[
+\[
   \begin{eqnarray*}
   Entities &=& \{id \mid entity(id,attrs) \in I\}\\
   Plans &=& \{pl \mid wasAssociatedWith(id,ag,act,pl,attrs) \in I, pl
   \neq -\}\\
-  Collections &=& \{e \mid memberOf(e',e) \in I\} \\
+  Collections &=& \{e \mid hadMember(e',e) \in I\} \\
   &\cup& \{e \mid
   entity(e,attrs) \in I, prov:type=prov:emptyCollection \in attrs\}\\
   
   Activities &=& \{id \mid activity(id,attrs) \in I\}\\
+  &\cup& \{a_{id},a'_{id} \mid id \in Entities\}\\
   &\cup& \{a_{id} \mid wasDerivedFrom(id,e_2,e_1,-,-,-,attrs) \in I\}\\
   Agents &=& \{id \mid agent(id,attrs) \in I\}\\
   \\
   Usages &=&  \{id \mid used(id,a,e,t,attrs) \in I\}\\
-  &\cup& \{u_{id} \mid wasDerivedFrom(id,e_2,e_1,-,-,-,attrs) \in I\}\\  Generations &=&  \{id \mid wasGeneratedBy(id,e,a,t,attrs) \in I\}\\
-  &\cup& \{g_{id} \mid wasDerivedFrom(id,e_2,e_1,-,-,-,attrs) \in I\}\\  Invalidations &=&  \{id \mid wasInvalidatedBy(id,e,a,t,attrs) \in I\}\\
+  &\cup& \{u_{id} \mid wasDerivedFrom(id,e_2,e_1,-,-,-,attrs) \in
+  I\}\\
+  Generations &=&  \{id \mid wasGeneratedBy(id,e,a,t,attrs) \in I\}\\
+  &\cup& \{g_{id} \mid wasDerivedFrom(id,e_2,e_1,-,-,-,attrs) \in
+  I\}\\
+ & \cup & \{g_{id} \mid id \in Entities\}\\
+   Invalidations &=&  \{id \mid wasInvalidatedBy(id,e,a,t,attrs) \in
+  I\}\\
+  & \cup & \{i_{id} \mid id \in Entities\}\\
   Starts &=&  \{id \mid wasStartedBy(id,a,e,a',t,attrs) \in I\}\\
   Ends &=&  \{id \mid wasEndedBy(id,a,e,a',t,attrs) \in I\}\\
   Events &=& Usages \cup Generations \cup Invalidations \cup Starts
@@ -3821,31 +3918,41 @@
 events in which $e$ participated.</p>
 \[
 \begin{eqnarray*}
-events(e) &=& \{id \mid used(id,a,e,t,attrs) \in I\}\\
+events'(e) &=& \{id \mid used(id,a,e,t,attrs) \in I\}\\
 &\cup& \{id \mid wasGeneratedBy(id,e,a,t,attrs) \in I\}\\
 &\cup& \{id \mid wasInvalidatedBy(id,e,a,t,attrs) \in I\}\\
 &\cup& \{id \mid wasStartedBy(id,a,e,a',t,attrs) \in I\}\\
 &\cup& \{id \mid wasEndedBy(id,a,e,a',t,attrs) \in I\}\\
-lifetime(e) &=& \{time(e) \mid evt \in events(e)\}\\
-value(e,a) &=& \{v \mid entity(e,attrs) \in I, a=v \in attrs\}\\
+&\cup& \{g_e,i_e\}\\
+events(e) &=& events'(e) \cup \bigcup_{specializationOf(e',e) \in I} events'(e')\\
+value'(e,a) &=& \{v \mid entity(e,attrs) \in I, (a=v) \in attrs\}
+\quad (a \neq uniq)\\
+value'(e,uniq) &=&\{ uniq_{e}\}\\
+value(e,a) &=&  value'(e) \cup \bigcup_{specializationOf(e',e) \in I} value'(e')\\
 thingOf(e) &=& [e]_\equiv
 \end{eqnarray*}
 \]
+<p>Above, we introduce a fresh attribute name $uniq$, not already in
+use in $I$, along with a fresh value $e$ and for each entity $e$ we
+add an attribute-value pair $uniq=uniq_e$ to $values(e,uniq)$.  This
+construction ensures that if an entity is a specialization of another
+in $I$ then the specialization relationship will hold in $M(I)$.  We
+also define the set of all events involved in $e$ as the set of events
+immediately involved in $e$ or any specialization of $e$.  Similarly,
+the values of attributes of $e$ are those immediately declared for $e$
+along with those of any specialization.  We also introduce dummy
+generation and invalidation events for each entity $e$, along with
+activities $a_e,a'_e$ to perform them.
+</p>
 <p> Similarly, for $Things$, we
 employ an auxiliary function $events:Things \to P(Events)$ that collects the set of all
 events in which one of the entities constituting the thing participated.</p>
 \[
 \begin{eqnarray*}
 events(T) &=& \bigcup_{e \in T} events(e)\\
-lifetime(e) &=& \{time(e) \mid evt \in events(T)\}\\
-value(T,a,t) &=& \bigcup_{e \in T, t \in lifetime(e)} value(e,a)\\
+value(T,a,evt) &=& \bigcup_{e \in T, evt \in events(e)} value(e,a)\\
 \end{eqnarray*}
 \]
-<div class="note">
-  <p> TODO: The above treatment of time/lifetime is flawed, as we are
-defining lifetimes to be sets of times, not intervals. What if
-  some of the times are symbolic?</p>
-  </div>
 
 <p> The functions $startTime$ and $endTime$ mapping activities to
   their start and end times is defined as follows:
@@ -3857,10 +3964,11 @@
   endTime(id) &=& t_2 \text{ where } activity(a,t_1,t_2,attrs) \in
   I\\
   & \text{ or } wasEndedBy(end,a,e,a',t_2,attrs) \in I\\
+  \end{eqnarray*}
 \]
   <p>Note that the above definition is deterministic because the start
   and end times in $activity$ and $wasStartedBy/wasEndedBy$ statements
-  must be equal</p>.
+  must be equal.</p>
 <p> The function $time$ mapping $Events$ to their $Times$ is defined
   as follows:
   </p>
@@ -3879,8 +3987,13 @@
   </p>
 \[\begin{eqnarray*}
   used(id) &=& (a,e) \text{ where } used(id,a,e,t,attrs) \in I\\
-  generated(id) &=&  (e,a) \text{ where } wasGeneratedBy(id,e,a,t,attrs) \in I\\
+  used(u_{id}) &=& (a_{id},e_1) \text{ where } wasDerivedFrom(id,e_2,e_1,-,-,-,attrs) \in I\\
+  generated(id) &=&  (e,a) \text{ where }
+  wasGeneratedBy(id,e,a,t,attrs) \in I\\
+ generated(g_{id}) &=& (e_2,a_{id}) \text{ where } wasDerivedFrom(id,e_2,e_1,-,-,-,attrs) \in I\\
+ generated(g_e) &=& (e,a_e) \text{ where } e \in Entities\\
   invalidated(id) &=& (e,a) \text{ where } wasInvalidatedBy(id,e,a,t,attrs) \in I\\
+ invalidated(i_e) &=& (e,a'_e) \text{ where } e \in Entities\\
   started(id) &=& (a,e,a') \text{ where } wasStartedBy(id,a,e,a',t,attrs) \in I\\
   ended(id) &=& (a,e,a') \text{ where }wasEndedBy(id,a,e,a',t,attrs) \in I\\
  \\
@@ -3888,7 +4001,7 @@
   I\\
  attributedTo(id) &=& (e,ag) \text{ where } wasAttributedTo(id,e,ag,attrs) \in I\\
 actedFor (id) &=& (ag_2,ag_1,act) \text{ where } actedOnBehalfOf(id,ag_2,ag_1,act,attrs) \in I\\
-  communicatedBy(id) &=& (a_2,a_1) \text{ where }
+  communicated(id) &=& (a_2,a_1) \text{ where }
   wasInformedBy(id,a_2,a_1,attrs)\in I\\
   derivationPath(id) &=& e_2\cdot g \cdot a \cdot u \cdot e_1 \text{ where } wasDerivedFrom(id,e_2,e_1,a,g,u,attrs) \in I\\
   derivationPath(id) &=& e_2\cdot g_{id} \cdot a_{id} \cdot u_{id} \cdot e_1 \text{ where } wasDerivedFrom(id,e_2,e_1,-,-,-,attrs) \in I\\
@@ -3899,13 +4012,7 @@
   imprecise derivations, we generate additional activities, generations
   and usages linking $e_2$ to $e_1$.
   </p>
-  <div class="remark">
-
-    <p> We explicitly add activities, generations, and usages to
-  ensure that we can form derivation paths for imprecise derivation
-  formulas.
-    </p>
-    </div>
+
 </section>
 <section>
 <h4>Relations</h4>
@@ -3918,9 +4025,9 @@
   for the directed graph that is used during validation of $I$ to
   test for cycles amond event ordering constraints.  See Sec. 7.1 of PROV-CONSTRAINTS [[PROV-CONSTRAINTS]].</p>
 
-    <p> Finally, the collection membership relation $MemberOf$ is
+    <p> Finally, the collection membership relation $Contains$ is
     defined as follows:</p>
-    \[(e,c) \in MemberOf \iff memberOf(e,c) \in I\]
+    \[(c,e) \in Contains \iff hadMember(c,e) \in I\]
     
 </section>
     <section>
@@ -3931,7 +4038,60 @@
     side-conditions in the components are satisfied.  As noted above,
     the disjointness constraints are satisfied by construction. </p>
 
-    </section>
+<p>For each axiom we give the corresponding justification:</p>
+
+<ol><li>
+  Axiom 1 follows because $I$ is normalized with respect to Inference 6.
+  </li>
+      <li> Axiom 2 follows from the construction, since we add dummy
+  generation and invalidation events for every entity.</li>
+<li>
+Axioms 3 and 4 follow because $I$ is normalized with respect to
+  Inference 9 and 10 respectively.
+  </li>
+  <li>Axiom 5 follows because $I$ is normalized with respect to
+  Inference 12.
+  </li>
+  <li>
+  Axioms 6 and 7 follow because $I$ is normalized with respect to
+  Inference 13 and 14 respectively.
+  </li><li>
+Axioms 8 through 17 follow because $I$ is normalized with respect to
+  Inference 15.
+  </li><li>
+Axioms 18 through 21 follow because $I$ is normalized with respect to
+  uniqueness constraints 24 through 27.
+  </li><li>
+  Axiom 22 follows because constraints 30, 31, 33, 34 ensure that a
+  start event for an activity precedes any other start, end, usage or
+  generation events involving that activity.
+  </li>
+  <li>
+  Axiom 23 follows because constraints 30, 32, 33, 34 ensure that an
+  end event for an activity follows any other events involving that activity.
+  </li>
+  <li>
+  Axiom 24 follows because constraints 34, 36, 37, 39 ensure that a
+  generation event for an entity precedes any other events involving that entity.
+  </li>
+  <li>
+  Axiom 25 follows because constraints 36, 38, 40, 43, 44 ensure that an
+  invalidation event for an entity  follows any other generation,
+  usage, or invalidation events involving
+  that entity.
+  </li>
+  <li>Axiom 26 follows from constraint 41.</li>
+  <li> Axiom 27 follows from constraint 42 and from the fact that the
+  event ordering constraint graph $G_I$ associated with a valid
+  instance $I$ cannot have any cycles involving a strict precedence
+  edge.
+  </li>
+  <li> Axioms 28 through 31 follow from Constraint 47.</li>
+  <li> Axioms 32 and 33 follow from Constraint 48.</li>
+  <li> Axioms 34 and 35 follow from Constraint 49.</li>
+
+  </ol>
+  </section>
     
 <section>
 <h4>Main results</h4>
@@ -3942,9 +4102,9 @@
     satisfies all of the inferences and constraints.  Thus, a form of
     completeness holds: every valid PROV instance has a model.</p>
 
-<div class="theorem">
+<div class="theorem" id="weak-completeness-theorem">
   <p>Suppose $J$ is a valid PROV instance and $I$ is a normal form for
-  $J$.  Then $M(I) \models J$</p>
+  $J$.  Then $M(I) \models J$.</p>
   </div>
   <div class="proof">
     <p>
@@ -3960,7 +4120,7 @@
     semantics and the definition of the construction of $M$.  </p>
 
     <p> To conclude, we need to show that if $J$ is not in normal
-    form, and norma,izes to $I$, then $M(I) \models J$.  WE can prove
+    form, and normalizes to $I$, then $M(I) \models J$.  We can prove
     this by induction on the length of the sequence of normalization
     steps.  The base case, when $J = I$, is established already.
     Suppose $J$ normalizes in $n+1$ steps and we can perform one
@@ -3981,14 +4141,14 @@
     attrs_2)\}$, where $S$ is a unifying substitution making $S(a_i) =
     S(b_i)$ for each $i \in \{1,\ldots,n\}$.  Since $M(I) \models J'$,
     we must have $M(I),\rho \models J'$  for some $\rho$, and therefore
-    we must also have that $M(I),\rho \models S(J_0)$ and $M(I)\rho \models r(id,S(a_1),\ldots,S(a_n),attrs_1\cup
+    we must also have that $M(I),\rho \models S(J_0)$ and $M(I),\rho \models r(id,S(a_1),\ldots,S(a_n),attrs_1\cup
     attrs_2)$.  We can extend $\rho$ to a valuation $\rho'$ such that
-    $M(I),\rho' \models S(x_1) = x_i \wedge \cdots \wedge S(x_k) =
-    x_k$ where $dom(S) = \{x_1,\ldots,x_k\}$.
+    $M(I),\rho' \models S(x_1) = x_1 \wedge \cdots \wedge S(x_k) =
+    x_k$ where $dom(S) = \{x_1,\ldots,x_k\}$.  Also, 
     $M(I),\rho' \models J_0$ and $M(I),\rho' \models r(id,a_1,\ldots,a_n,attrs_1\cup
     attrs_2)$.  Moreover, since $S$ is a unifier, we also have $M(I),\rho' \models r(id,b_1,\ldots,b_n,attrs_1\cup
     attrs_2)$.  Finally, since we can always remove attributes from an
-    atomic formula without danaging its satisfiability, we can
+    atomic formula without damaging its satisfiability, we can
     conclude that $M(I),\rho' \models r(id,a_1,\ldots,a_n,attrs_1)
     \wedge r(id,b_1,\ldots,b_n,    attrs_2)$.  To conclude, we have
     shown $M(I) \models J_0 \cup \{ r(id,a_1,\ldots,a_n,attrs_1)
--- a/semantics/releases/NOTE-prov-sem-20130430/Overview.html	Fri Apr 05 18:11:45 2013 +0100
+++ b/semantics/releases/NOTE-prov-sem-20130430/Overview.html	Fri Apr 05 18:13:06 2013 +0100
@@ -735,45 +735,6 @@
 .section dl.attrs dd, .section dl.eldef dd {
     margin-bottom:  0;
 }
-</style><style>/* --- ISSUES/NOTES --- */
-div.issue-title, div.note-title {
-    padding-right:  1em;
-    min-width: 7.5em;
-    color: #b9ab2d;
-}
-div.issue-title { color: #e05252; }
-div.note-title { color: #52e052; }
-div.issue-title span, div.note-title span {
-    text-transform: uppercase;
-}
-div.note, div.issue {
-    margin-top: 1em;
-    margin-bottom: 1em;
-}
-.note > p:first-child, .issue > p:first-child { margin-top: 0 }
-.issue, .note {
-    padding: .5em;
-    border-left-width: .5em;
-    border-left-style: solid;
-}
-div.issue, div.note {
-    padding: 0.5em;
-    margin: 1em 0;
-    position: relative;
-    clear: both;
-}
-span.note, span.issue { padding: .1em .5em .15em; }
-
-.issue {
-    border-color: #e05252;
-    background: #fbe9e9;
-}
-.note {
-    border-color: #52e052;
-    background: #e9fbe9;
-}
-
-
 </style><link rel="stylesheet" href="http://www.w3.org/StyleSheets/TR/W3C-WG-NOTE"><style type="text/css">DIV.MathJax_MathML {text-align: center; margin: .75em 0px}
 .MathJax_MathML {font-style: normal; font-weight: normal; line-height: normal; font-size: 100%; font-size-adjust: none; text-indent: 0; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; border: 0; padding: 0; margin: 0}
 span.MathJax_MathML {display: inline}
@@ -815,6 +776,13 @@
   </dl>
   
   
+    <p>
+      
+        This document is also available in this non-normative format: 
+      
+      <a rel="alternate" href="prov-sem.pdf">PDF</a>
+    </p>
+  
   
   
     
@@ -849,8 +817,9 @@
 data model (called the <dfn id="dfn-naive-semantics">naive semantics</dfn>), viewing
 PROV-DM statements as atomic formulas in the sense of first-order
 logic, and viewing the constraints and inferences specified in
-PROV-CONSTRAINTS as a first-order theory. It is shown that the
-first-order theory is sound with respect to the naive semantics.
+PROV-CONSTRAINTS as a first-order theory. It is shown that valid PROv
+instances (in the sense of PROV-CONSTRAINTS) correspond to satisfiable
+theories.
 This information may be useful to researchers or users of PROV to
 understand the intended meaning and use of PROV for modeling
 information about the actual history, derivation or evolution of Web
@@ -876,22 +845,22 @@
 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. 
 <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/PR-prov-constraints-20130312/">PROV-CONSTRAINTS</a>
-(Proposed Recommendation), a set of constraints applying to the PROV
+<li> <a href="http://www.w3.org/TR/2013/NOTE-prov-overview-20130430/">PROV-OVERVIEW</a> (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/NOTE-prov-primer-20130430/">PROV-PRIMER</a> (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/REC-prov-o-20130430/">PROV-O</a> (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/REC-prov-dm-20130430/">PROV-DM</a> (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/REC-prov-n-20130430/">PROV-N</a> (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/REC-prov-constraints-20130430/">PROV-CONSTRAINTS</a>
+(Recommendation), a set of constraints applying to the PROV
 data model [<cite><a class="bibref" href="#bib-PROV-CONSTRAINTS">PROV-CONSTRAINTS</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 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
+<li> <a href="http://www.w3.org/TR/2013/NOTE-prov-xml-20130430/">PROV-XML</a> (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/NOTE-prov-aq-20130430/">PROV-AQ</a> (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/NOTE-prov-dictionary-20130430/">PROV-DICTIONARY</a> (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/NOTE-prov-dc-20130430/">PROV-DC</a> (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/NOTE-prov-sem-20130430/">PROV-SEM</a>
+(Note), a declarative specification in terms of
 first-order logic of the PROV data model (this document);</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/NOTE-prov-links-20130430/">PROV-LINKS</a> (Note) introduces a mechanism to link across bundles [<cite><a class="bibref" href="#bib-PROV-LINKS">PROV-LINKS</a></cite>].</li>
 </ul>
 
 
@@ -939,7 +908,7 @@
       
     
   
-</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" class="tocxref"><span class="secno">2.2 </span> Times </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><li class="tocline"><a href="#collections-1" class="tocxref"><span class="secno">3.2.1.2 </span>Collections</a></li></ul></li><li class="tocline"><a href="#activities-1" class="tocxref"><span class="secno">3.2.2 </span> Activities </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="#influences-1" class="tocxref"><span class="secno">3.2.4 </span> Influences </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="#attributions-1" class="tocxref"><span class="secno">3.2.4.3 </span> Attributions </a></li><li class="tocline"><a href="#communications-1" class="tocxref"><span class="secno">3.2.4.4 </span>Communications</a></li><li class="tocline"><a href="#delegations-1" class="tocxref"><span class="secno">3.2.4.5 </span>Delegations</a></li><li class="tocline"><a href="#derivations-1" class="tocxref"><span class="secno">3.2.4.6 </span> Derivations </a></li></ul></li></ul></li><li class="tocline"><a href="#additional-axioms" class="tocxref"><span class="secno">3.3 </span>Additional axioms</a></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" class="tocxref"><span class="secno">4.4.5 </span> Start </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="#delegation" class="tocxref"><span class="secno">4.4.9 </span> Delegation </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="#semantics-of-auxiliary-formulas" class="tocxref"><span class="secno">4.5 </span>Semantics of Auxiliary Formulas</a><ul class="toc"><li class="tocline"><a href="#precedes-and-strictly-precedes" class="tocxref"><span class="secno">4.5.1 </span>Precedes and Strictly Precedes</a></li><li class="tocline"><a href="#notnull" class="tocxref"><span class="secno">4.5.2 </span>notNull</a></li><li class="tocline"><a href="#typeof" class="tocxref"><span class="secno">4.5.3 </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-completeness" class="tocxref"><span class="secno">6. </span>Soundness and Completeness</a><ul class="toc"><li class="tocline"><a href="#soundness" class="tocxref"><span class="secno">6.1 </span>Soundness</a></li><li class="tocline"><a href="#completeness" class="tocxref"><span class="secno">6.2 </span>Weak Completeness</a><ul class="toc"><li class="tocline"><a href="#sets" class="tocxref"><span class="secno">6.2.1 </span>Sets</a></li><li class="tocline"><a href="#functions" class="tocxref"><span class="secno">6.2.2 </span>Functions</a></li><li class="tocline"><a href="#relations" class="tocxref"><span class="secno">6.2.3 </span>Relations</a></li><li class="tocline"><a href="#axioms-1" class="tocxref"><span class="secno">6.2.4 </span>Axioms</a></li><li class="tocline"><a href="#main-results" class="tocxref"><span class="secno">6.2.5 </span>Main results</a></li></ul></li></ul></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 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="#attributes-and-values" class="tocxref"><span class="secno">2.2 </span> Attributes and Values </a></li><li class="tocline"><a href="#times" class="tocxref"><span class="secno">2.3 </span> Times </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></li><li class="tocline"><a href="#objects-1" class="tocxref"><span class="secno">4. </span> Objects </a><ul class="toc"><li class="tocline"><a href="#entities-1" class="tocxref"><span class="secno">4.1 </span> Entities </a><ul class="toc"><li class="tocline"><a href="#plans-1" class="tocxref"><span class="secno">4.1.1 </span> Plans </a></li><li class="tocline"><a href="#collections-1" class="tocxref"><span class="secno">4.1.2 </span>Collections</a></li></ul></li><li class="tocline"><a href="#activities-1" class="tocxref"><span class="secno">4.2 </span> Activities </a></li><li class="tocline"><a href="#agents-1" class="tocxref"><span class="secno">4.3 </span> Agents </a></li><li class="tocline"><a href="#influences-1" class="tocxref"><span class="secno">4.4 </span> Influences </a><ul class="toc"><li class="tocline"><a href="#events-1" class="tocxref"><span class="secno">4.4.1 </span> Events </a></li><li class="tocline"><a href="#associations-1" class="tocxref"><span class="secno">4.4.2 </span> Associations </a></li><li class="tocline"><a href="#attributions-1" class="tocxref"><span class="secno">4.4.3 </span> Attributions </a></li><li class="tocline"><a href="#communications-1" class="tocxref"><span class="secno">4.4.4 </span>Communications</a></li><li class="tocline"><a href="#delegations-1" class="tocxref"><span class="secno">4.4.5 </span>Delegations</a></li><li class="tocline"><a href="#derivations-1" class="tocxref"><span class="secno">4.4.6 </span> Derivations </a></li></ul></li></ul></li><li class="tocline"><a href="#additional-axioms" class="tocxref"><span class="secno">5. </span>Additional axioms</a></li><li class="tocline"><a href="#putting-it-all-together" class="tocxref"><span class="secno">6. </span> Putting it all together </a></li><li class="tocline"><a href="#interpretations" class="tocxref"><span class="secno">7. </span> Interpretations </a></li><li class="tocline"><a href="#semantics" class="tocxref"><span class="secno">8. </span> Semantics </a><ul class="toc"><li class="tocline"><a href="#satisfaction" class="tocxref"><span class="secno">8.1 </span> Satisfaction </a></li><li class="tocline"><a href="#attribute-matching" class="tocxref"><span class="secno">8.2 </span> Attribute matching </a></li><li class="tocline"><a href="#semantics-of-element-formulas" class="tocxref"><span class="secno">8.3 </span> Semantics of Element Formulas </a><ul class="toc"><li class="tocline"><a href="#entity" class="tocxref"><span class="secno">8.3.1 </span> Entity </a></li><li class="tocline"><a href="#activity" class="tocxref"><span class="secno">8.3.2 </span> Activity </a></li><li class="tocline"><a href="#agent" class="tocxref"><span class="secno">8.3.3 </span> Agent </a></li></ul></li><li class="tocline"><a href="#semantics-of-relations" class="tocxref"><span class="secno">8.4 </span> Semantics of Relations </a><ul class="toc"><li class="tocline"><a href="#generation" class="tocxref"><span class="secno">8.4.1 </span> Generation </a></li><li class="tocline"><a href="#use" class="tocxref"><span class="secno">8.4.2 </span> Use </a></li><li class="tocline"><a href="#invalidation" class="tocxref"><span class="secno">8.4.3 </span> Invalidation </a></li><li class="tocline"><a href="#association" class="tocxref"><span class="secno">8.4.4 </span> Association </a></li><li class="tocline"><a href="#start" class="tocxref"><span class="secno">8.4.5 </span> Start </a></li><li class="tocline"><a href="#end" class="tocxref"><span class="secno">8.4.6 </span> End </a></li><li class="tocline"><a href="#attribution" class="tocxref"><span class="secno">8.4.7 </span> Attribution </a></li><li class="tocline"><a href="#communication" class="tocxref"><span class="secno">8.4.8 </span>Communication</a></li><li class="tocline"><a href="#delegation" class="tocxref"><span class="secno">8.4.9 </span> Delegation </a></li><li class="tocline"><a href="#derivation" class="tocxref"><span class="secno">8.4.10 </span> Derivation </a><ul class="toc"><li class="tocline"><a href="#precise" class="tocxref"><span class="secno">8.4.10.1 </span> Precise </a></li><li class="tocline"><a href="#imprecise" class="tocxref"><span class="secno">8.4.10.2 </span> Imprecise </a></li></ul></li><li class="tocline"><a href="#influence" class="tocxref"><span class="secno">8.4.11 </span>Influence</a></li><li class="tocline"><a href="#specialization" class="tocxref"><span class="secno">8.4.12 </span> Specialization </a></li><li class="tocline"><a href="#alternate" class="tocxref"><span class="secno">8.4.13 </span> Alternate </a></li><li class="tocline"><a href="#membership" class="tocxref"><span class="secno">8.4.14 </span> Membership </a></li></ul></li><li class="tocline"><a href="#semantics-of-auxiliary-formulas" class="tocxref"><span class="secno">8.5 </span>Semantics of Auxiliary Formulas</a><ul class="toc"><li class="tocline"><a href="#precedes-and-strictly-precedes" class="tocxref"><span class="secno">8.5.1 </span>Precedes and Strictly Precedes</a></li><li class="tocline"><a href="#notnull" class="tocxref"><span class="secno">8.5.2 </span>notNull</a></li><li class="tocline"><a href="#typeof" class="tocxref"><span class="secno">8.5.3 </span>typeOf</a></li></ul></li></ul></li><li class="tocline"><a href="#theory" class="tocxref"><span class="secno">9. </span> Inferences and Constraints </a><ul class="toc"><li class="tocline"><a href="#inferences" class="tocxref"><span class="secno">9.1 </span>Inferences</a></li><li class="tocline"><a href="#constraints" class="tocxref"><span class="secno">9.2 </span>Constraints</a><ul class="toc"><li class="tocline"><a href="#uniqueness-constraints" class="tocxref"><span class="secno">9.2.1 </span>Uniqueness constraints</a></li><li class="tocline"><a href="#ordering-constraints" class="tocxref"><span class="secno">9.2.2 </span>Ordering constraints</a></li><li class="tocline"><a href="#typing-constraints" class="tocxref"><span class="secno">9.2.3 </span>Typing constraints</a></li><li class="tocline"><a href="#impossibility-constraints" class="tocxref"><span class="secno">9.2.4 </span>Impossibility constraints</a></li></ul></li></ul></li><li class="tocline"><a href="#soundness-completeness" class="tocxref"><span class="secno">10. </span>Soundness and Completeness</a><ul class="toc"><li class="tocline"><a href="#soundness" class="tocxref"><span class="secno">10.1 </span>Soundness</a></li><li class="tocline"><a href="#completeness" class="tocxref"><span class="secno">10.2 </span>Weak Completeness</a><ul class="toc"><li class="tocline"><a href="#sets" class="tocxref"><span class="secno">10.2.1 </span>Sets</a></li><li class="tocline"><a href="#functions" class="tocxref"><span class="secno">10.2.2 </span>Functions</a></li><li class="tocline"><a href="#relations" class="tocxref"><span class="secno">10.2.3 </span>Relations</a></li><li class="tocline"><a href="#axioms-1" class="tocxref"><span class="secno">10.2.4 </span>Axioms</a></li><li class="tocline"><a href="#main-results" class="tocxref"><span class="secno">10.2.5 </span>Main results</a></li></ul></li></ul></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>
 
 
 
@@ -983,7 +952,7 @@
 <h3><span class="secno">1.1 </span>Purpose of this document</h3>
 
 </section>
-<p>The PROV-DM and PROV-CONSTRAINTS give motivating examples that
+<p>The PROV-DM and PROV-CONSTRAINTS specifications give motivating examples that
 provide an intuition about the meaning of the constructs.  For some
 concepts, such as use, start, end, generation/invalidation, and
 derivation, the meaning is either obvious or situation-dependent.
@@ -1006,13 +975,10 @@
 intent behind certain features of PROV, to researchers investigating
 richer forms of reasoning over provenance, or to future efforts
 building upon PROV.  It is intended as an exploration of <b>one</b> semantics for PROV, not a definitive specification of the  <b>only</b>
-semantics of PROV.  We intend to provide an intuitive semantics that satisfies all
-of the constraints on valid PROV instances, which ensures that no
-invalid PROV instance has a model.  The current naive semantics, however, is
-not complete in the sense that some valid PROV instances lack models.</p>
-<div class="note"><div class="note-title"><span>Note</span></div><div class="">
-  <p>TODO: Revise this to reflect future improvements in the semantics.</p>
-  </div></div>
+semantics of PROV.  We provide a semantics that satisfies all
+of the constraints on valid PROV instances, and such that valid PROV
+instances correspond to satisfiable theories: every valid instance has
+a model, and vice versa.</p>
 
 <p> Although it is a work in progress, the naive semantics has some appealing
 properties.  Specifically, it provides a declarative counterpart to
@@ -1022,7 +988,7 @@
 to keep the specification closer to implementations, although other
 implementations are possible and allowed.  In addition to providing a
 naive semantics, this document shows that the operational
-presentation of PROV validity checking is sound with respect to the
+presentation of PROV validity checking is equivalent to the
 declarative presentation adopted here.  This could help justify
 alternative approaches to validity checking.</p>
 
@@ -1061,10 +1027,7 @@
   including soundness and a weak form of completeness: a PROV
   instance is valid if and only if it has a naive model.</li>
 </ul>
-<div class="note"><div class="note-title"><span>Note</span></div><div class="">
-<p>TODO: We would like to say something stronger here, such as a
-  completeness result for naive models, but this will take more work.</p>
-  </div></div>
+
 
 </section>
 <section id="audience">
@@ -1077,7 +1040,7 @@
 [<cite><a class="bibref" href="#bib-PROV-N">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.
+  presented in <a href="http://www.w3.org/TR/2013/REC-prov-constraints-20130430/#overview">Section 2.5</a> of PROV-CONSTRAINTS.
 </p>
 
   <p>This document may be useful to users of PROV who have a formal
@@ -1151,15 +1114,25 @@
 </p>
 </section>
 
+<section id="attributes-and-values">
+<h3><span class="secno">2.2 </span> Attributes and Values </h3>
+
+<p>We assume a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-7-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-7">Attributes</script> of attribute labels and a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-8-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-8">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_MathML" id="MathJax-Element-9-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-9">P(Value)</script>, that is, the set of sets of values.
+</p> </section>
+
+
 
 <section id="times">
-<h3><span class="secno">2.2 </span> Times </h3>
-
-<p>We assume an ordered set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-7-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi><mo stretchy="false">,</mo><mo stretchy="false">≤</mo><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-7">(Times,\leq)</script> of time instants, where
-<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-8-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>V</mi><mi>a</mi><mi>l</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-8">Times \subseteq Val</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-9-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">≤</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-9">\leq</script> is a linear order.
+<h3><span class="secno">2.3 </span> Times </h3>
+
+<p>We assume an ordered set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-10-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi><mo stretchy="false">,</mo><mo stretchy="false">≤</mo><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-10">(Times,\leq)</script> of time instants, where
+<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-11-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>V</mi><mi>a</mi><mi>l</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-11">Times \subseteq Val</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-12-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">≤</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-12">\leq</script> is a linear order.
 </p>
 
-
+<!--
 <div class="remark">
 <p>Restricting attention to linearly-ordered times, and imposing this
   order on events, is a simplifying assumption; it is more restrictive than required to model the
@@ -1170,12 +1143,12 @@
   </p>
   </div>
   
-<p>We also consider a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-10-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-10">Intervals</script> of subsets of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-11-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-11">Times</script>  of the
-  form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-12-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><mi>t</mi><mo stretchy="false">∣</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">≤</mo><mi>t</mi><mo stretchy="false">≤</mo><msub><mi>t</mi><mn>2</mn></msub><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-12">[t_1,t_2] = \{t \mid t_1 \leq t \leq t_2\}</script>.  Every interval has a minimum
-  and maximum time, written <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-13-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">=</mo><msub><mi>t</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-13">min([t_1,t_2]) = t_1</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-14-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">=</mo><msub><mi>t</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-14">max([t_1,t_2])
-  = t_2</script> respectively.
+<p>We also consider a set $Intervals$ of subsets of $Times$  of the
+  form $[t_1,t_2] = \{t \mid t_1 \leq t \leq t_2\}$.  Every interval has a minimum
+  and maximum time, written $min([t_1,t_2]) = t_1$ and $max([t_1,t_2])
+  = t_2$ respectively.
 </p>
-
+-->
   
 </section>
   
@@ -1183,26 +1156,17 @@
 
 
   
-<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_MathML" id="MathJax-Element-15-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-15">Attributes</script> of attribute labels and a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-16-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-16">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_MathML" id="MathJax-Element-17-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-17">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_MathML" id="MathJax-Element-18-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-18">r(id,a_1,\ldots,a_n)</script> instead of the semicolon notation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-19-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">;</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-19">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 class="MathJax_MathML" id="MathJax-Element-13-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-13">r(id,a_1,\ldots,a_n)</script> instead of the semicolon notation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-14-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">;</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-14">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 "-".
-  Placeholder symbols <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-20-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">−</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-20">-</script> can only appear in the specified arguments
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-21-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>l</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-21">pl</script> in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-22-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-22">wasAssociatedWith</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-23-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo stretchy="false">,</mo><mi>g</mi><mo stretchy="false">,</mo><mi>u</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-23">a,g,u</script> in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-24-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mi>a</mi><mi>s</mi><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>r</mi><mi>o</mi><mi>m</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-24">wasDerivedFrom</script>, as shown in the grammar below.
+  Placeholder symbols <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-15-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">−</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-15">-</script> can only appear in the specified arguments
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-16-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>l</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-16">pl</script> in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-17-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-17">wasAssociatedWith</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-18-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo stretchy="false">,</mo><mi>g</mi><mo stretchy="false">,</mo><mi>u</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-18">a,g,u</script> in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-19-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mi>a</mi><mi>s</mi><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>r</mi><mi>o</mi><mi>m</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-19">wasDerivedFrom</script>, as shown in the grammar below.
 </p>
-<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-25-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi>a</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>i</mi><mi>c</mi><mi mathvariant="normal">_</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>u</mi><mi>l</mi><mi>a</mi></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">::=</mo></mrow></mtd><mtd><mi>e</mi><mi>l</mi><mi>e</mi><mi>m</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi mathvariant="normal">_</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>u</mi><mi>l</mi><mi>a</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>r</mi><mi>e</mi><mi>l</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi mathvariant="normal">_</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>u</mi><mi>l</mi><mi>a</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>a</mi><mi>u</mi><mi>x</mi><mi>i</mi><mi>l</mi><mi>i</mi><mi>a</mi><mi>r</mi><mi>y</mi><mi mathvariant="normal">_</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>u</mi><mi>l</mi><mi>a</mi></mtd></mtr><mtr><mtd><mi>e</mi><mi>l</mi><mi>e</mi><mi>m</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi mathvariant="normal">_</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>u</mi><mi>l</mi><mi>a</mi></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">::=</mo></mrow></mtd><mtd><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>a</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>s</mi><mi>t</mi><mo stretchy="false">,</mo><mi>e</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>r</mi><mi>e</mi><mi>l</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi mathvariant="normal">_</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>u</mi><mi>l</mi><mi>a</mi></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">::=</mo></mrow></mtd><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>S</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>E</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>T</mi><mi>o</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>a</mi><mi>c</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>O</mi><mi>n</mi><mi>B</mi><mi>e</mi><mi>h</mi><mi>a</mi><mi>l</mi><mi>f</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mi>f</mi><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>r</mi><mi>o</mi><mi>m</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>g</mi><mo stretchy="false">,</mo><mi>u</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>r</mi><mi>o</mi><mi>m</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>x</mi><mo stretchy="false">,</mo><mi>y</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>h</mi><mi>a</mi><mi>d</mi><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>a</mi><mi>u</mi><mi>x</mi><mi>i</mi><mi>l</mi><mi>i</mi><mi>a</mi><mi>r</mi><mi>y</mi><mi mathvariant="normal">_</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>u</mi><mi>l</mi><mi>a</mi></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">::=</mo></mrow></mtd><mtd><mi>x</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">strictlyPrecedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>y</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>x</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>y</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">,</mo><mi>t</mi><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi></mtd><mtd><mo stretchy="false">::=</mo></mtd><mtd><mo stretchy="false">[</mo><mi>a</mi><mi>t</mi><mi>t</mi><msub><mi>r</mi><mn>1</mn></msub><mo stretchy="false">=</mo><msub><mi>v</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><msub><mi>r</mi><mi>n</mi></msub><mo stretchy="false">=</mo><msub><mi>v</mi><mi>n</mi></msub><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mi>t</mi><mi>y</mi></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">::=</mo></mrow></mtd><mtd><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>a</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>y</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo 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 \newcommand{\precedes}{~\mathrel{\textrm{precedes}}~}
 \newcommand{\strictlyPrecedes}{~\mathrel{\textrm{strictlyPrecedes}}~}
   \begin{array}{rcl}
@@ -1245,28 +1209,28 @@
 </script>
 
 <div class="remark">
-  <p>We include the standard PROV collection types (<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-26-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-26">Collection</script> and
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-27-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-27">EmptyCollection</script>) and the membership relation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-28-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mi>a</mi><mi>d</mi><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-28">hadMember</script>; however,
+  <p>We include the standard PROV collection types (<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-21-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-21">Collection</script> and
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-22-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-22">EmptyCollection</script>) and the membership relation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-23-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mi>a</mi><mi>d</mi><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-23">hadMember</script>; however,
   we do not model dictionaries or the insertion or deletion relations
   in PROV-DICTIONARY [<cite><a class="bibref" href="#bib-PROV-DICTIONARY">PROV-DICTIONARY</a></cite>], since these are not part
   of the PROV recommendations.  If these features are incorporated
   into future standards, their semantics (and the soundness of the
   associated constraints) should be modeled.
-  We omit the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-29-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>r</mi><mi>o</mi><mi>v</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-29">prov</script> prefixes from the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-30-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-30">Collection</script> and
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-31-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-31">EmptyCollection</script> types.
+  We omit the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-24-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>r</mi><mi>o</mi><mi>v</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-24">prov</script> prefixes from the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-25-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-25">Collection</script> and
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-26-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-26">EmptyCollection</script> types.
   </p>
 
   <p>As stated in the Introduction, we do not explicitly model
   bundles or PROV documents; however, each instance can be viewed as
   a set of formulas and can be modeled separately.  The semantics of
   the standard features of PROV can be defined without talking about
-  multiple instances; however, the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-32-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-32">mentionOf</script> relation in
+  multiple instances; however, the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-27-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-27">mentionOf</script> relation in
   PROV-LINKS [<cite><a class="bibref" href="#bib-PROV-LINKS">PROV-LINKS</a></cite>] is intended to support linking across
-  bundles.  Future editions of PROV may incorporate <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-33-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-33">mentionOf</script> or
+  bundles.  Future editions of PROV may incorporate <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-28-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-28">mentionOf</script> or
   other cross-instance assertions, and if so this semantics should be
   generalized in order to provide a rationale for such an
-  extension and to establish the soundness of constaints associated
-  with <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-34-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-34">mentionOf</script>. </p>
+  extension and to establish the soundness of constraints associated
+  with <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-29-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-29">mentionOf</script>. </p>
 </div>
 </section>
 
@@ -1275,7 +1239,7 @@
 
 <p>We also consider the usual connectives and quantifiers of
 first-order logic [<cite><a class="bibref" href="#bib-Logic">Logic</a></cite>].</p>
-<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-35-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="italic">ϕ</mi></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">::=</mo></mrow></mtd><mtd><mi>a</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>i</mi><mi>c</mi><mi mathvariant="normal">_</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>u</mi><mi>l</mi><mi>a</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>T</mi><mi>r</mi><mi>u</mi><mi>e</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>x</mi><mo stretchy="false">=</mo><mi>y</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi mathvariant="normal">¬</mi><mtext>&nbsp;</mtext><mi mathvariant="italic">ϕ</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><msub><mi mathvariant="italic">ϕ</mi><mn>1</mn></msub><mo stretchy="false">∧</mo><msub><mi mathvariant="italic">ϕ</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><msub><mi mathvariant="italic">ϕ</mi><mn>1</mn></msub><mo stretchy="false">∨</mo><msub><mi mathvariant="italic">ϕ</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><msub><mi mathvariant="italic">ϕ</mi><mn>1</mn></msub><mo stretchy="false">⇒</mo><msub><mi mathvariant="italic">ϕ</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi mathvariant="normal">∀</mi><mi>x</mi><mo>.</mo><mi mathvariant="italic">ϕ</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi mathvariant="normal">∃</mi><mi>x</mi><mo>.</mo><mi mathvariant="italic">ϕ</mi></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-35">
+<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-30-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="italic">ϕ</mi></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">::=</mo></mrow></mtd><mtd><mi>a</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>i</mi><mi>c</mi><mi mathvariant="normal">_</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>u</mi><mi>l</mi><mi>a</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>T</mi><mi>r</mi><mi>u</mi><mi>e</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>x</mi><mo stretchy="false">=</mo><mi>y</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi mathvariant="normal">¬</mi><mtext>&nbsp;</mtext><mi mathvariant="italic">ϕ</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><msub><mi mathvariant="italic">ϕ</mi><mn>1</mn></msub><mo stretchy="false">∧</mo><msub><mi mathvariant="italic">ϕ</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><msub><mi mathvariant="italic">ϕ</mi><mn>1</mn></msub><mo stretchy="false">∨</mo><msub><mi mathvariant="italic">ϕ</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><msub><mi mathvariant="italic">ϕ</mi><mn>1</mn></msub><mo stretchy="false">⇒</mo><msub><mi mathvariant="italic">ϕ</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi mathvariant="normal">∀</mi><mi>x</mi><mo>.</mo><mi mathvariant="italic">ϕ</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi mathvariant="normal">∃</mi><mi>x</mi><mo>.</mo><mi mathvariant="italic">ϕ</mi></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-30">
 \begin{array}{rcl}
   \phi &{::=}& atomic\_formula\\
 & | & True\\
@@ -1310,115 +1274,125 @@
 <section id="structures">
 <!--OddPage--><h2><span class="secno">3. </span> Structures and Interpretations </h2>
 
-<p> In this section we define mathematical structures <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-36-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-36">W</script> that can be used to
+<p> In this section we define mathematical structures <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-31-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-31">W</script> that can be used to
 interpret PROV formulas and instances.  A structure consists of a
 collection of sets, functions and relations.  The components of a
-structure <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-37-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-37">W</script> are given in the rest of the section in
+structure <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-32-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-32">W</script> are given in the rest of the section in
 <em>components</em>, highlighted in boxes.
-</p><section id="things-1">
-
-<h3><span class="secno">3.1 </span> Things </h3> 
-
-<p><em>Things</em>  is a set of things in the situation being modeled.  Each thing has a lifetime during which it exists and attributes whose values can change over time.
+
+
+
+</p><h2 id="things-1"> Things </h2> 
+
+<p><em>Things</em>  is a set of things in the situation being modeled.
+Each thing has an associates set of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-33-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-33">Events</script> and attributes whose
+values can change over time.  Different kinds of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-34-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-34">Events</script> are specified further below.
 </p>
-<p>To model this, a structure <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-38-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-38">W</script> includes:
+<p>To model this, a structure <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-35-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-35">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_MathML" id="MathJax-Element-39-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-39">Things</script> of things</li>
-  <li> a function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-40-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">:</mo><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi><mo stretchy="false">→</mo><mi>I</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-40">lifetime : Things \to Intervals</script> from things to intervals</li>
-  <li>a function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-41-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">:</mo><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi><mo stretchy="false">→</mo><mi>P</mi><mo stretchy="false">(</mo><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-41">value : Things \times Attributes \times Times \to P(Values)</script>
+  <li> a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-36-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-36">Things</script> of things</li>
+  <li> a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-37-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-37">Events</script> of events</li>
+  <li> a function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-38-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">:</mo><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi><mo stretchy="false">→</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-38">events : Things \to P(Events)</script> from things to
+  sets of events.</li>
+  <li>a function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-39-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">:</mo><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">→</mo><mi>P</mi><mo stretchy="false">(</mo><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-39">value : Things \times Attributes \times Events \to
+  P(Values)</script> giving the possible values of each attribute of a
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-40-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-40">Thing</script> at the instant of a given event.
 </li>
-</ol>
+<li>Attributes are only defined during the events of a thing, that
+is, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-41-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">≠</mo><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-41">value(T,a,evt) \neq \emptyset</script> implies <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-42-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-42">evt \in events(T)</script>.
+</li></ol>
 </div>
 <p>
-The range of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-42-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-42">value</script> is the set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-43-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-43">P(Values)</script>, indicating that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-44-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-44">value</script>
-is essentially a multi-valued function that returns a set of values (possibly empty).    When <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-45-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-45">value(x,a,t) =
-\emptyset</script>, we say that attribute <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-46-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-46">a</script> is undefined for <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-47-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-47">x</script> at time <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-48-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-48">t</script>.</p>
+The range of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-43-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-43">value</script> is the set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-44-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-44">P(Values)</script>, indicating that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-45-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-45">value</script>
+is essentially a multi-valued function that returns a set of values (possibly empty).    When <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-46-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-46">value(x,a,evt) =
+\emptyset</script>, we say that attribute <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-47-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-47">a</script> is undefined for <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-48-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-48">x</script> at event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-49-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-49">evt</script>.</p>
 
 <p>Note that this description does not say what the structure of a
-<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-49-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-49">Thing</script> is, only how it may be described in terms of its lifetime
+<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-50-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-50">Thing</script> is, only how it may be described in terms of its events
 and attribute values.  A thing could 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_MathML" id="MathJax-Element-50-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="italic">π</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-50">\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_MathML" id="MathJax-Element-51-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-51">Thing</script> to its time interval and attribute mapping.
+it could be a transcendental number like <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-51-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="italic">π</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-51">\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_MathML" id="MathJax-Element-52-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-52">Thing</script> to its events and attribute mapping.
 </p>
 
 
 <p>The identity of a Thing is not observable through its attributes or
-lifetime, so it is possible for two different <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-52-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-52">Things</script> to be indistinguishable by their
-attribute values and lifetime.  That is, if the set of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-53-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>T</mi><mn>0</mn></msub><mo stretchy="false">,</mo><msub><mi>T</mi><mn>1</mn></msub><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-53">Things = \{T_0,T_1\}</script> and the attributes are
-specified as <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-54-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><msub><mi>T</mi><mn>0</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><msub><mi>T</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-54">value(T_0,a,t) = value(T_1,a,t)</script> for each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-55-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo stretchy="false">∈</mo><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-55">t\in
-Times</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-56-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo stretchy="false">∈</mo><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-56">a \in Attributes</script>, this does not imply that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-57-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>0</mn></msub><mo stretchy="false">=</mo><msub><mi>T</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-57">T_0 = T_1</script>.
+events, so it is possible for two different <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-53-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-53">Things</script> to be indistinguishable by their
+attribute values and events.  That is, if the set of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-54-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>T</mi><mn>0</mn></msub><mo stretchy="false">,</mo><msub><mi>T</mi><mn>1</mn></msub><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-54">Things = \{T_0,T_1\}</script> and the attributes are
+specified as <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-55-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><msub><mi>T</mi><mn>0</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><msub><mi>T</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-55">value(T_0,a,evt) = value(T_1,a,evt)</script> for each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-56-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-56">evt\in
+Events</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-57-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo stretchy="false">∈</mo><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-57">a \in Attributes</script>, this does not imply that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-58-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>0</mn></msub><mo stretchy="false">=</mo><msub><mi>T</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-58">T_0 = T_1</script>.
 </p>
 
 
 </section>
 
 <section id="objects-1">
-<h3><span class="secno">3.2 </span> Objects </h3>
-
-
-<p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-58-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-58">Things</script> are things in the world that have attributes that
-  can change over time.  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-59-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-59">Things</script> may not have distinguishing features
+<!--OddPage--><h2><span class="secno">4. </span> Objects </h2>
+
+
+<p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-59-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-59">Things</script> are things in the world that have attributes that
+  can change over time.  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-60-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-60">Things</script> may not have distinguishing features
   that are readily observable and permanent.  In PROV, we do not talk
-  explicitly about <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-60-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-60">Things</script>, but instead we talk about various objects
+  explicitly about <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-61-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-61">Things</script>, but instead we talk about various objects
   that have discrete, fixed features,  and relationships among these
-  objects. Some objects, called <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-61-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-61">Entities</script>, are associated with
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-62-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-62">Things</script>, and their fixed attributes need to match those of the
-  associated <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-63-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-63">Thing</script> during their common lifetime.  Others correspond
+  objects. Some objects, called <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-62-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-62">Entities</script>, are associated with
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-63-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-63">Things</script>, and their fixed attributes need to match those of the
+  associated <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-64-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-64">Thing</script> during their common events.  Others correspond
   to agents, activities, or identifiable interactions among them.</p>  
 
-<p>In this section, we detail the different subsets of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-64-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-64">Objects</script>, and
+<p>In this section, we detail the different subsets of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-65-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-65">Objects</script>, and
 give disjointness constraints and associated functions.  Generally, these constraints are necessary to validate
 disjointness constraints from PROV-CONSTRAINTS [<cite><a class="bibref" href="#bib-PROV-CONSTRAINTS">PROV-CONSTRAINTS</a></cite>].
 </p>
 
 <p>
-An <em>Object</em> is described by a time interval and attributes with
+An <em>Object</em> is described by a set of events and attributes with
 fixed values.  Objects encompass entities, activities, agents, and
 interactions (i.e., usage, generation, and other events or influence relations).
 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_MathML" id="MathJax-Element-65-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-65">Objects</script> 
-</li><li> a function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-66-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">:</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi><mo stretchy="false">→</mo><mi>I</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-66">lifetime : Objects \to Intervals</script> from objects to time intervals
-</li><li> a function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-67-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">:</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>s</mi><mo stretchy="false">→</mo><mi>P</mi><mo stretchy="false">(</mo><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-67">value : Objects \times Attributes \to P(Values)</script>
+  <ol><li> a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-66-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-66">Objects</script> 
+</li><li> a function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-67-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">:</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi><mo stretchy="false">→</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-67">events : Objects \to P(Events)</script> from objects
+  to associated sets of events.
+</li><li> a function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-68-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">:</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>s</mi><mo stretchy="false">→</mo><mi>P</mi><mo stretchy="false">(</mo><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-68">value : Objects \times Attributes \to P(Values)</script>.
 </li></ol>
 </div>
 
-<p>Intuitively, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-68-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-68">lifetime(e)</script> is the time interval during which object
-<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-69-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-69">e</script> exists.  The set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-70-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-70">value(e,a)</script> is the set of values of attribute <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-71-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-71">a</script> during the object's lifetime.
+<p>Intuitively, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-69-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-69">events(e)</script> is the set of events in which <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-70-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-70">e</script> participated.  The set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-71-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-71">value(e,a)</script> is the set of values of attribute <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-72-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-72">a</script> during the object's events.
 </p>
 
-<p>As with <em>Things</em>, the range of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-72-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-72">value</script> is sets of values,
-making <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-73-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-73">value</script> effectively a multivalued function.  It is also
+<p>As with <em>Things</em>, the range of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-73-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-73">value</script> is sets of values,
+making <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-74-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-74">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_MathML" id="MathJax-Element-74-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-74">Objects</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-75-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-75">Things</script> are disjoint; however, certain objects,
+sets of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-75-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-75">Objects</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-76-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-76">Things</script> are disjoint; however, certain objects,
 namely entities, are associated with things.
 </p>
 
 <div class="remark">
   <p>
-  Disjointness between <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-76-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-76">Objects</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-77-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-77">Things</script> is not necessary but is
+  Disjointness between <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-77-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-77">Objects</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-78-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-78">Things</script> is not necessary but is
   assumed in order to avoid confusion between the different categories
-  (time-varying <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-78-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-78">Things</script> vs fixed <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-79-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-79">Objects</script>).
+  (time-varying <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-79-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-79">Things</script> vs fixed <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-80-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-80">Objects</script>).
   </p>
   </div>
   
 <section id="entities-1">
-<h4><span class="secno">3.2.1 </span> Entities </h4>
+<h3><span class="secno">4.1 </span> Entities </h3>
 
 <p>An <em>entity</em> is a kind of object that fixes some aspects of a
   thing. We assume:</p>
 
 <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_MathML" id="MathJax-Element-80-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-80">Entities \subseteq Objects</script> of entities, disjoint from <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-81-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-81">Activities</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-82-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-82">Events</script> below.
+  <ol><li> a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-81-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-81">Entities \subseteq Objects</script> of entities, disjoint from <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-82-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-82">Activities</script> below.
 </li><li> a function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-83-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">:</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">→</mo><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-83">thingOf : Entities \to Things</script> that associates
-  each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-84-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-84">Entity</script> <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-85-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-85">e</script> with a <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-86-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-86">Thing</script>, such that for each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-87-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo stretchy="false">∈</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-87">t \in
-  lifetime(e)</script>, and for each attribute <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-88-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-88">a</script> we have <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-89-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">⊆</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-89">value(e,a)
-  \subseteq value(thingOf(e),a,t)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-90-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">⊆</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-90">lifetime(e) \subseteq lifetime(thingOf(e))</script>.
+  each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-84-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-84">Entity</script> <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-85-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-85">e</script> with a <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-86-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-86">Thing</script>, such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-87-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">⊆</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-87">events(e) \subseteq
+  events(thingOf(e))</script> and for each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-88-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-88">evt \in
+  events(e)</script> and for each attribute <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-89-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-89">a</script> we have <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-90-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">⊆</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-90">value(e,a)
+  \subseteq value(thingOf(e),a,evt)</script>.  
 </li>
 <!--<li>a relation $SpecializationOf \subseteq Entities \times Entities$
   that is irreflexive and transitive.  Furthermore, if $(e_1,e_2) \in
@@ -1426,7 +1400,7 @@
 <ol><li>
   $thingOf(e_1) = thingOf(e_2)$
   </li>
-  <li>$lifetime(e_1) \subseteq lifetime(e_2)$</li>
+  <li>$events(e_1) \subseteq events(e_2)$</li>
   <li>For each attribute $attr$ we have $value(e_1,attr) \supseteq
   value(e_2,attr)$.</li>
   </ol></li>
@@ -1437,19 +1411,19 @@
 
 <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_MathML" id="MathJax-Element-91-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-91">value(x,a,t) = \emptyset</script> means that the
-  attribute <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-92-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-92">a</script> has no value at time <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-93-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-93">t</script>, whereas for an entity,
+  different: for a thing, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-91-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-91">value(x,a,evt) = \emptyset</script> means that the
+  attribute <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-92-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-92">a</script> has no value at event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-93-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-93">evt</script>, whereas for an entity,
   <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-94-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-94">value(x,a) = \emptyset</script> only means that the thing associated to
   entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-95-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-95">x</script> need not have a
-  fixed value for <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-96-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-96">a</script> during the lifetime of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-97-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-97">x</script>.  This does not imply
-  that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-98-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-98">value(thingOf(e),a,t) = \emptyset</script> when <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-99-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo stretchy="false">∈</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-99">t \in lifetime(e)</script>.
+  fixed value for <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-96-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-96">a</script> during the events of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-97-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-97">x</script>.  This does not imply
+  that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-98-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-98">value(thingOf(e),a,evt) = \emptyset</script> when <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-99-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-99">evt \in events(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_MathML" id="MathJax-Element-100-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-100">value(thingOf(e),a,t)</script> is <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-101-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-101">\{1\}</script> at
-  some time in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-102-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-102">lifetime(e)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-103-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><mn>2</mn><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-103">value(thingOf(e),a,t') = \{2\}</script> at
-  some other time <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-104-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>t</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-104">t'</script>.  Then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-105-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-105">value(e,a)</script> must be <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-106-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-106">\emptyset</script> because
+  be present in the associated thing throughout the events of the
+  entity.  For example, suppose <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-100-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-100">value(thingOf(e),a,evt)</script> is <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-101-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-101">\{1\}</script> at
+  some event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-102-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-102"> evt \in events(e)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-103-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</mi><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><mn>2</mn><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-103">value(thingOf(e),a,evt') = \{2\}</script> at
+  some other event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-104-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><msup><mi>t</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-104">evt'</script>.  Then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-105-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-105">value(e,a)</script> must be <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-106-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-106">\emptyset</script> because
   there is no other set of values that is simultaneously contained in
   both <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-107-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-107">\{1\}</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-108-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>2</mn><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-108">\{2\}</script>.  </p> </div>
 
@@ -1458,11 +1432,11 @@
 <div class="remark">
   <p>
   In the above description of how <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-109-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-109">Entities</script> relate to <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-110-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-110">Things</script>, we
-  require  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-111-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">⊆</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-111">value(e,a) \subseteq
-  value(thingOf(e),a,t)</script> whenever <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-112-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo stretchy="false">∈</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-112">t \in lifetime(e)</script>.  Intuitively, this means that if we are
+  require  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-111-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">⊆</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-111">value(e,a) \subseteq
+  value(thingOf(e),a,evt)</script> whenever <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-112-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-112">evt \in events(e)</script>.  Intuitively, this means that if we are
   talking about a <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-113-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-113">Thing</script> indirectly by describing an <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-114-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-114">Entity</script>, then
   any attributes we ascribe to the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-115-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-115">Entity</script> must also describe the
-  associated <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-116-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-116">Thing</script> during their common lifetime.  Attributes of both
+  associated <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-116-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-116">Thing</script> during their common events.  Attributes of both
   <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-117-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-117">Entities</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-118-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-118">Things</script> are multi-valued, so there is no
   inconsistency in saying that an entity has two different values for
   some attribute.  In some
@@ -1481,7 +1455,7 @@
   </div>
   
 <section id="plans-1">  
-<h5><span class="secno">3.2.1.1 </span> Plans </h5>
+<h4><span class="secno">4.1.1 </span> Plans </h4>
 <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>
@@ -1490,12 +1464,12 @@
 </section>
 
 <section id="collections-1">
-  <h5><span class="secno">3.2.1.2 </span>Collections</h5>
+  <h4><span class="secno">4.1.2 </span>Collections</h4>
   <p>We identify another specific subset of the entities called
   <em>collections</em>, with the following associated structure:</p>
   <div class="component" id="collections" data-count="5" data-title="Component 5 (collections)"><div class="ruleTitle"><a class="internalDFN" href="#collections">Component 5 (collections)</a></div>
     <ul><li>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-128-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-128">Collections \subseteq Entities</script></li>
-    <li>A membership relation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-129-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>O</mi><mi>f</mi><mo stretchy="false">⊆</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-129">MemberOf\subseteq Entities \times Collections</script>
+    <li>A membership relation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-129-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>n</mi><mi>t</mi><mi>a</mi><mi>i</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">×</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-129">Contains\subseteq Collections \times Entities</script>
   indicating when an entity is a member of another (collection)
   entity.</li>
   </ul>
@@ -1504,7 +1478,7 @@
   </section>
 
     <section id="activities-1">
-<h4><span class="secno">3.2.2 </span> Activities </h4>
+<h3><span class="secno">4.2 </span> Activities </h3>
 
 
 <p>An <em>activity</em> is an object that encompasses a set of events.  We introduce:
@@ -1519,7 +1493,7 @@
 </div></section>
   
   <section id="agents-1">
-<h4><span class="secno">3.2.3 </span> Agents </h4>
+<h3><span class="secno">4.3 </span> Agents </h3>
 
 <p>An agent is an object that can act, by controlling, starting,
   ending, or participating in activities.  An agent is something that
@@ -1539,156 +1513,156 @@
 
 
 <section id="influences-1">
-<h4><span class="secno">3.2.4 </span> Influences </h4>
+<h3><span class="secno">4.4 </span> Influences </h3>
 
 <p>We consider a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-135-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-135">Influences \subseteq Objects</script> which has disjoint
   subsets
-  <em>Events</em> connecting entities and activities,
-  <em>Associations</em> between agents and activities,
-    <em>Attributions</em> between entities and agents,
-  <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
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-136-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-136">Events</script> connecting entities and activities,
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-137-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-137">Associations</script> between agents and activities,
+    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-138-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-138">Attributions</script> between entities and agents,
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-139-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-139">Communications</script> between pairs of activities,
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-140-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-140">Delegations</script> between pairs of agents, and
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-141-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-141">Derivations</script> that describe chains of generation and usage
   steps.  These kinds of influences are discussed further below.  Influences are disjoint from entities, activities and agents.
 </p>
 <div class="component" id="influences" data-count="8" data-title="Component 8 (influences)"><div class="ruleTitle"><a class="internalDFN" href="#influences">Component 8 (influences)</a></div>
-  <ol><li> A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-136-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi><mo stretchy="false">=</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>D</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-136">Influences = Events \cup Associations \cup
+  <ol><li> A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-142-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi><mo stretchy="false">=</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>D</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-142">Influences = Events \cup Associations \cup
   Communications \cup Delegations \cup Derivations \subseteq Objects</script>
 </li>
-<li> The sets <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-137-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-137">Events</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-138-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-138">Associations</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-139-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-139">Communications</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-140-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-140">Delegations</script>
-  and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-141-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-141">Derivations</script> are all pairwise disjoint.
+<li> The sets <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-143-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-143">Events</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-144-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-144">Associations</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-145-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-145">Communications</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-146-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-146">Delegations</script>
+  and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-147-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-147">Derivations</script> are all pairwise disjoint.
 </li><li> Influences are disjoint from entities, agents and
-activities:  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-142-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi><mo stretchy="false">∩</mo><mo stretchy="false">(</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>A</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-142">Influences \cap (Entities \cup Activities \cup Agents) = \emptyset</script>
+activities:  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-148-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi><mo stretchy="false">∩</mo><mo stretchy="false">(</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>A</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-148">Influences \cap (Entities \cup Activities \cup Agents) = \emptyset</script>
 </li>
-<li>An associated function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-143-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">:</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi><mo stretchy="false">→</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi><mo stretchy="false">×</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-143">influenced : Influences \to
+<li>An associated function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-149-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">:</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi><mo stretchy="false">→</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi><mo stretchy="false">×</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-149">influenced : Influences \to
   Objects \times Objects</script> giving the source and target of each influence.</li>
 </ol>
 </div>
 
 
 <section id="events-1">
-<h5><span class="secno">3.2.4.1 </span> Events </h5>
-
-<p>An <em>Event</em> is an influence whose lifetime is a single time
+<h4><span class="secno">4.4.1 </span> Events </h4>
+
+<p>An <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-150-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-150">Event</script> is an influence whose events is a single time
 instant, and relates an activity to an entity (which could be an
 agent).  Events have types including usage, generation, invalidation, starting and ending.  Events are instantaneous.  We introduce:
 </p>
 <div class="component" id="events" data-count="9" data-title="Component 9 (events)"><div class="ruleTitle"><a class="internalDFN" href="#events">Component 9 (events)</a></div>
-<ol><li> A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-144-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-144">Events \subseteq Influences</script> of events, partitioned
-  into disjoint subsets <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-145-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>s</mi><mo stretchy="false">,</mo><mi>E</mi><mi>n</mi><mi>d</mi><mi>s</mi><mo stretchy="false">,</mo><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">,</mo><mi>U</mi><mi>s</mi><mi>a</mi><mi>g</mi><mi>e</mi><mi>s</mi><mo stretchy="false">,</mo><mi>I</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-145">Starts, Ends, Generations, Usages,
+<ol><li> A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-151-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-151">Events \subseteq Influences</script> of events, partitioned
+  into disjoint subsets <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-152-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>s</mi><mo stretchy="false">,</mo><mi>E</mi><mi>n</mi><mi>d</mi><mi>s</mi><mo stretchy="false">,</mo><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">,</mo><mi>U</mi><mi>s</mi><mi>a</mi><mi>g</mi><mi>e</mi><mi>s</mi><mo stretchy="false">,</mo><mi>I</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-152">Starts, Ends, Generations, Usages,
   Invalidations</script>.
-</li><li> A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-146-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">:</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">→</mo><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-146">time : Events \to Times</script> giving the time of each
-event, such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-147-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-147">lifetime(evt) = \{time(evt)\}</script>.
+</li><li> A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-153-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">:</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">→</mo><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-153">time : Events \to Times</script>.
 </li>
-<li> A quasi-ordering on events <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-148-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">⪯⊂</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">×</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-148">\preceq \subset Events \times
-Events</script>.  We write <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-149-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo stretchy="false">≺</mo><msup><mi>e</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-149">e \prec e'</script> when <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-150-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo stretchy="false">⪯</mo><msup><mi>e</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-150">e \preceq e'</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-151-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">⪯̸</mo><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-151">e'
+<li> A quasi-ordering on events <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-154-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">⪯⊂</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">×</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-154">\preceq \subset Events \times
+Events</script>.  We write <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-155-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo stretchy="false">≺</mo><msup><mi>e</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-155">e \prec e'</script> when <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-156-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo stretchy="false">⪯</mo><msup><mi>e</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-156">e \preceq e'</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-157-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">⪯̸</mo><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-157">e'
 \not\preceq e</script> hold.
 </li>
-<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-152-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">:</mo><mi>S</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>s</mi><mo stretchy="false">→</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-152">started : Starts \to Activities \times Entities \times Activities</script>.
-</li>
-<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-153-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">:</mo><mi>E</mi><mi>n</mi><mi>d</mi><mi>s</mi><mo stretchy="false">→</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-153">ended : Ends \to Activities \times Entities \times Activities</script>.
+<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-158-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">:</mo><mi>S</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>s</mi><mo stretchy="false">→</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-158">started : Starts \to Activities \times Entities \times Activities</script>.
 </li>
-<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-154-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">:</mo><mi>U</mi><mi>s</mi><mi>a</mi><mi>g</mi><mi>e</mi><mi>s</mi><mo stretchy="false">→</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-154">used : Usages \to Activities \times Entities</script>.
+<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-159-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">:</mo><mi>E</mi><mi>n</mi><mi>d</mi><mi>s</mi><mo stretchy="false">→</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-159">ended : Ends \to Activities \times Entities \times Activities</script>.
 </li>
-<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-155-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">:</mo><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-155">generated : Generations \to Entities \times Activities</script>.
+<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-160-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">:</mo><mi>U</mi><mi>s</mi><mi>a</mi><mi>g</mi><mi>e</mi><mi>s</mi><mo stretchy="false">→</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-160">used : Usages \to Activities \times Entities</script>.
 </li>
-<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-156-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">:</mo><mi>I</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-156">invalidated : Invalidations \to Entities \times Activities</script>.
+<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-161-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">:</mo><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-161">generated : Generations \to Entities \times Activities</script>.
+</li>
+<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-162-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">:</mo><mi>I</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-162">invalidated : Invalidations \to Entities \times Activities</script>.
 </li>
 </ol>
 </div>
 </section>
 <section id="associations-1">
 
-<h5><span class="secno">3.2.4.2 </span> Associations </h5>
-
-<p>An <em>Association</em> is an influence relating an agent to an activity.  To model associations, we introduce:
+<h4><span class="secno">4.4.2 </span> Associations </h4>
+
+<p>An <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-163-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-163">Association</script> is an influence relating an agent to an activity
+and optional plan.  To model associations, we introduce:
 </p>
 <div class="component" id="associations" data-count="10" data-title="Component 10 (associations)"><div class="ruleTitle"><a class="internalDFN" href="#associations">Component 10 (associations)</a></div>
-  <p>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-157-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-157">Associations \subseteq Influences</script> with associated
-  function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-158-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">:</mo><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>A</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>P</mi><mi>l</mi><mi>a</mi><mi>n</mi><msub><mi>s</mi><mi mathvariant="normal">⊥</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-158">associatedWith : Associations \to  Agents \times Activities \times Plans_\bot</script>.
+  <p>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-164-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-164">Associations \subseteq Influences</script> with associated
+  function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-165-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">:</mo><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>A</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>P</mi><mi>l</mi><mi>a</mi><mi>n</mi><msub><mi>s</mi><mi mathvariant="normal">⊥</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-165">associatedWith : Associations \to  Agents \times Activities \times Plans_\bot</script>.
 </p>
   </div>
   </section>
 <section id="attributions-1">
 
-<h5><span class="secno">3.2.4.3 </span> Attributions </h5>
-
-<p>An <em>Attribution</em> is an influence relating an entity to an agent.  To model associations, we introduce:
+<h4><span class="secno">4.4.3 </span> Attributions </h4>
+
+<p>An <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-166-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-166">Attribution</script> is an influence relating an entity to an agent.  To model attributions, we introduce:
 </p>
 <div class="component" id="attributions" data-count="11" data-title="Component 11 (attributions)"><div class="ruleTitle"><a class="internalDFN" href="#attributions">Component 11 (attributions)</a></div>
-  <p>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-159-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-159">Attributions \subseteq Influences</script> with associated
-  function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-160-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>T</mi><mi>o</mi><mo stretchy="false">:</mo><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-160">attributedTo : Attributions \to Entities \times Agents</script>.
+  <p>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-167-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-167">Attributions \subseteq Influences</script> with associated
+  function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-168-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>T</mi><mi>o</mi><mo stretchy="false">:</mo><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-168">attributedTo : Attributions \to Entities \times Agents</script>.
 </p>
   </div>
   
 </section>
   <section id="communications-1">
-  <h5><span class="secno">3.2.4.4 </span>Communications</h5>
-  <p>A <em>Communication</em> is an influence indicating exchange of
+  <h4><span class="secno">4.4.4 </span>Communications</h4>
+  <p>A <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-169-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-169">Communication</script> is an influence indicating exchange of
   information between activities.  To model communications, we introduce:
 </p>
 <div class="component" id="communications" data-count="12" data-title="Component 12 (communications)"><div class="ruleTitle"><a class="internalDFN" href="#communications">Component 12 (communications)</a></div>
-  <p>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-161-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-161">Communications \subseteq Influences</script> with associated
-  function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-162-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">:</mo><mi>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-162">communicatedBy : Communications \to Activities \times Activities</script>.
+  <p>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-170-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-170">Communications \subseteq Influences</script> with associated
+  function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-171-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">:</mo><mi>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-171">communicated : Communications \to Activities \times Activities</script>.
 </p>
   </div>
   
 
 </section>
   <section id="delegations-1">
-  <h5><span class="secno">3.2.4.5 </span>Delegations</h5>
-<p>A <em>Delegation</em> is an influence relating  two agents.  To
+  <h4><span class="secno">4.4.5 </span>Delegations</h4>
+<p>A <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-172-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-172">Delegation</script> is an influence relating  two agents.  To
   model delegations, we introduce:
 </p>
 <div class="component" id="delegations" data-count="13" data-title="Component 13 (delegations)"><div class="ruleTitle"><a class="internalDFN" href="#delegations">Component 13 (delegations)</a></div>
-  <p>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-163-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-163">Delegations \subseteq Influences</script> and associated function
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-164-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>c</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>o</mi><mi>r</mi><mo stretchy="false">:</mo><mi>D</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>A</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-164">actedFor : Delegations \to Agents \times Agents \times Activities</script>
+  <p>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-173-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-173">Delegations \subseteq Influences</script> and associated function
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-174-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>c</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>o</mi><mi>r</mi><mo stretchy="false">:</mo><mi>D</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>A</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-174">actedFor : Delegations \to Agents \times Agents \times Activities</script>
 </p>
   </div>
   
 </section>
   <section id="derivations-1">
   
-  <h5><span class="secno">3.2.4.6 </span> Derivations </h5>
-
-<p>A <em>Derivation</em> is an influence chaining one or more
+  <h4><span class="secno">4.4.6 </span> Derivations </h4>
+
+<p>A <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-175-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-175">Derivation</script> is an influence chaining one or more
   generation and use steps.  To model derivations, we introduce an
   auxiliary notion of <em>derivation path</em>.  These paths are of the form </p>
 
-<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-165-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>e</mi><mi>n</mi><msub><mi>t</mi><mi>n</mi></msub><mo stretchy="false">⋅</mo><msub><mi>g</mi><mi>n</mi></msub><mo stretchy="false">⋅</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mi>n</mi></msub><mo stretchy="false">⋅</mo><msub><mi>u</mi><mi>n</mi></msub><mo stretchy="false">⋅</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo stretchy="false">−</mo><mn>1</mn></mrow></msub><mo stretchy="false">⋅</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">⋅</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">⋅</mo><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">⋅</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">⋅</mo><msub><mi>u</mi><mn>1</mn></msub><mo stretchy="false">⋅</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>0</mn></msub></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-165">ent_n\cdot g_n\cdot  act_n\cdot  u_n\cdot  ent_{n-1}\cdot  ...\cdot
+<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-176-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>e</mi><mi>n</mi><msub><mi>t</mi><mi>n</mi></msub><mo stretchy="false">⋅</mo><msub><mi>g</mi><mi>n</mi></msub><mo stretchy="false">⋅</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mi>n</mi></msub><mo stretchy="false">⋅</mo><msub><mi>u</mi><mi>n</mi></msub><mo stretchy="false">⋅</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo stretchy="false">−</mo><mn>1</mn></mrow></msub><mo stretchy="false">⋅</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">⋅</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">⋅</mo><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">⋅</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">⋅</mo><msub><mi>u</mi><mn>1</mn></msub><mo stretchy="false">⋅</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>0</mn></msub></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-176">ent_n\cdot g_n\cdot  act_n\cdot  u_n\cdot  ent_{n-1}\cdot  ...\cdot
 ent_1\cdot  g_1\cdot  act_1\cdot  u_1\cdot  ent_0</script>
 
-<p>where the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-166-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><msub><mi>t</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-166">ent_i</script> are entities, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-167-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>c</mi><msub><mi>t</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-167">act_i</script> are activities, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-168-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>g</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-168">g_i</script> are generations, and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-169-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-169">u_i</script> are usages.
+<p>where the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-177-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><msub><mi>t</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-177">ent_i</script> are entities, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-178-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>c</mi><msub><mi>t</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-178">act_i</script> are activities, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-179-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>g</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-179">g_i</script> are generations, and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-180-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-180">u_i</script> are usages.
 </p>
 <p>Formally, we consider the (regular) language:
 </p>
-<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-170-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mi>s</mi><mo stretchy="false">=</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">⋅</mo><mo stretchy="false">(</mo><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⋅</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">⋅</mo><mi>U</mi><mi>s</mi><mi>a</mi><mi>g</mi><mi>e</mi><mi>s</mi><mo stretchy="false">⋅</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><msup><mo stretchy="false">)</mo><mo stretchy="false">+</mo></msup></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-170">DerivationPaths = Entities \cdot (Generations \cdot Activities \cdot
+<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-181-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mi>s</mi><mo stretchy="false">=</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">⋅</mo><mo stretchy="false">(</mo><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⋅</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">⋅</mo><mi>U</mi><mi>s</mi><mi>a</mi><mi>g</mi><mi>e</mi><mi>s</mi><mo stretchy="false">⋅</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><msup><mo stretchy="false">)</mo><mo stretchy="false">+</mo></msup></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-181">DerivationPaths = Entities \cdot (Generations \cdot Activities \cdot
 Usages \cdot Entities)^+</script>
 <p>with the constraints that for each derivation path:
 </p>
 <ul>
-<li>for each substring <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-171-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">⋅</mo><mi>g</mi><mo stretchy="false">⋅</mo><mi>a</mi><mi>c</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-171">ent\cdot g \cdot act</script> we have <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-172-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-172">generated(g) = (ent,act)</script>, and
+<li>for each substring <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-182-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">⋅</mo><mi>g</mi><mo stretchy="false">⋅</mo><mi>a</mi><mi>c</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-182">ent\cdot g \cdot act</script> we have <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-183-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-183">generated(g) = (ent,act)</script>, and
 </li>
-<li>for each substring <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-173-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">⋅</mo><mi>u</mi><mo stretchy="false">⋅</mo><mi>e</mi><mi>n</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-173">act \cdot u \cdot ent</script> we have <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-174-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-174">used(u) = (act,ent)</script>.
+<li>for each substring <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-184-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">⋅</mo><mi>u</mi><mo stretchy="false">⋅</mo><mi>e</mi><mi>n</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-184">act \cdot u \cdot ent</script> we have <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-185-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-185">used(u) = (act,ent)</script>.
 </li>
 </ul>
 
 
 <div class="component" id="derivations" data-count="14" data-title="Component 14 (derivations)"><div class="ruleTitle"><a class="internalDFN" href="#derivations">Component 14 (derivations)</a></div>
-<p>  A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-175-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-175">Derivations \subseteq Influences</script> with an associated
-  function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-176-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mo stretchy="false">:</mo><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-176">derivationPath : Derivations \to
+<p>  A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-186-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-186">Derivations \subseteq Influences</script> with an associated
+  function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-187-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mo stretchy="false">:</mo><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-187">derivationPath : Derivations \to
 DerivationPaths</script>  linking each derivation to a derivation path.  </p>
 <p></p>
 </div>
 
 <div class="remark">
   <p>
-  The <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-177-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-177">derivationPath</script> function links each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-178-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo stretchy="false">∈</mo><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-178"> d \in Derivations</script> to a
+  The <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-188-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-188">derivationPath</script> function links each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-189-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo stretchy="false">∈</mo><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-189"> d \in Derivations</script> to a
   derivation path.  A derivation has exactly one associated derivation
   path.  However, if the PROV-N statement <span class="name">wasDerivedFrom(e_2,e_1,-,-,-)</span> is asserted in an
-  instance, there may be multiple derivation paths linking <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-179-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-179">e_2</script> to
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-180-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-180">e_1</script>, each corresponding to a different path, identified by different
-  derivations <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-181-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo stretchy="false">∈</mo><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-181">d \in Derivations</script>.
+  instance, there may be multiple derivation paths linking <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-190-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-190">e_2</script> to
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-191-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-191">e_1</script>, each corresponding to a different path, identified by different
+  derivations <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-192-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo stretchy="false">∈</mo><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-192">d \in Derivations</script>.
   </p>
 
   <p>A derivation path implies the existence of at least one chained generation
@@ -1701,7 +1675,7 @@
   </p>
 <p>
   The reason why we need paths and not just individual derivation
-  steps is to reflect that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-182-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mi>a</mi><mi>s</mi><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>r</mi><mi>o</mi><mi>m</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-182">wasDerivedFrom(id,e_2,e_1,-,-,-,attrs)</script> formulas can
+  steps is to reflect that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-193-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mi>a</mi><mi>s</mi><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>r</mi><mi>o</mi><mi>m</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-193">wasDerivedFrom(id,e_2,e_1,-,-,-,attrs)</script> formulas can
   represent multiple derivation steps.  However, there is no way to
   express a multi-step derivation path in PROV: any valid PROV
   instance turns out to have a model in which all derivation paths are
@@ -1712,7 +1686,7 @@
 </section>
 
   <section id="additional-axioms">
-  <h3><span class="secno">3.3 </span>Additional axioms</h3>
+  <!--OddPage--><h2><span class="secno">5. </span>Additional axioms</h2>
 
   <p> Above we have stated some properties of the components.  We
   impose some additional properties that relate several components, as
@@ -1722,96 +1696,136 @@
     
   <ol>
     <li id="axiom1">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-183-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-183">generated(g) = (e,a_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-184-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-184">used(u) = (a_2,e)</script> then there
-    exists <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-185-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo stretchy="false">∈</mo><mi>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-185">c \in Communications</script> such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-186-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-186">communicatedBy(c) = (a_2,a_1)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-194-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-194">generated(g) = (e,a_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-195-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-195">used(u) = (a_2,e)</script> then there
+    exists <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-196-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo stretchy="false">∈</mo><mi>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-196">c \in Communications</script> such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-197-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-197">communicated(c) = (a_2,a_1)</script>.
     </li>
     <li id="axiom2">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-187-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-187">started(start) = (a_2,e,a_1)</script> then there exists <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-188-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-188">gen</script> such
-    that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-189-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-189">generated(gen) = (e,a_1)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-198-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo stretchy="false">∈</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-198">e \in Entities</script> then there exist <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-199-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>i</mi><mi>n</mi><mi>v</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-199">gen,inv,a,a'</script> such that
+    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-200-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-200">generated(gen) = (e,a)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-201-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>n</mi><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-201">invalidated(inv) = (e,a')</script>.
     </li>
     <li id="axiom3">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-190-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-190">ended(end) = (a_2,e,a_1)</script> then there exists <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-191-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-191">gen</script> such
-    that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-192-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-192">generated(gen) = (e,a_1)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-202-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-202">started(start) = (a_2,e,a_1)</script> then there exists <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-203-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-203">gen</script> such
+    that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-204-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-204">generated(gen) = (e,a_1)</script>.
     </li>
     <li id="axiom4">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-193-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo stretchy="false">∈</mo><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-193">d \in Derivations</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-194-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>r</mi><mi>o</mi><mi>v</mi><mo stretchy="false">:</mo><mi>R</mi><mi>e</mi><mi>v</mi><mi>i</mi><mi>s</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">∈</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">,</mo><mi>p</mi><mi>r</mi><mi>o</mi><mi>v</mi><mo stretchy="false">:</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-194">prov:Revision \in
-    value(d,prov:type)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-195-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">⋅</mo><mi>w</mi><mo stretchy="false">⋅</mo><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-195">derivationPath(deriv) = e_2 \cdot w \cdot
-    e_1</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-196-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-196">thingOf(e_1) = thingOf(e_2)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-205-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-205">ended(end) = (a_2,e,a_1)</script> then there exists <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-206-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-206">gen</script> such
+    that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-207-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-207">generated(gen) = (e,a_1)</script>.
     </li>
     <li id="axiom5">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-197-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>T</mi><mi>o</mi><mo stretchy="false">(</mo><mi>a</mi><mi>t</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-197">attributedTo(att) = (e,ag)</script> then there exist <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-198-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-198">gen</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-199-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-199">assoc</script>
-    such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-200-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-200">generated(gen) = (e,a)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-201-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-201">associatedWith(assoc) = (a,ag)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-208-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo stretchy="false">∈</mo><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-208">d \in Derivations</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-209-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>r</mi><mi>o</mi><mi>v</mi><mo stretchy="false">:</mo><mi>R</mi><mi>e</mi><mi>v</mi><mi>i</mi><mi>s</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">∈</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">,</mo><mi>p</mi><mi>r</mi><mi>o</mi><mi>v</mi><mo stretchy="false">:</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-209">prov:Revision \in
+    value(d,prov:type)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-210-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">⋅</mo><mi>w</mi><mo stretchy="false">⋅</mo><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-210">derivationPath(deriv) = e_2 \cdot w \cdot
+    e_1</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-211-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-211">thingOf(e_1) = thingOf(e_2)</script>.
     </li>
     <li id="axiom6">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-202-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>c</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>o</mi><mi>r</mi><mo stretchy="false">(</mo><mi>d</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-202">actedFor(deleg) = (ag_2,ag_1,act)</script> then there exist
-    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-203-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><msub><mi>c</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><msub><mi>c</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-203">assoc_1,assoc_2,pl_1,pl_2</script> such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-204-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><msub><mi>c</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-204">associatedWith(assoc_1) = (act,ag_1,pl_1)</script>
-    and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-205-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><msub><mi>c</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-205">associatedWith(assoc_2) = (act,ag_2,pl_2)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-212-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>T</mi><mi>o</mi><mo stretchy="false">(</mo><mi>a</mi><mi>t</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-212">attributedTo(att) = (e,ag)</script> then there exist <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-213-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-213">gen</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-214-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-214">assoc</script>
+    such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-215-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-215">generated(gen) = (e,a)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-216-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-216">associatedWith(assoc) = (a,ag)</script>.
     </li>
     <li id="axiom7">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-206-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-206">generated(id) = (e,a)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-207-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-207">influenced(id) = (e,a)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-217-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>c</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>o</mi><mi>r</mi><mo stretchy="false">(</mo><mi>d</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-217">actedFor(deleg) = (ag_2,ag_1,act)</script> then there exist
+    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-218-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><msub><mi>c</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><msub><mi>c</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-218">assoc_1,assoc_2,pl_1,pl_2</script> such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-219-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><msub><mi>c</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-219">associatedWith(assoc_1) = (ag_1,act,pl_1)</script>
+    and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-220-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><msub><mi>c</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-220">associatedWith(assoc_2) = (ag_2,act,pl_2)</script>.
     </li>
     <li id="axiom8">
-        If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-208-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-208">used(id) = (e,a)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-209-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-209">influenced(id) = (e,a)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-221-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-221">generated(id) = (e,a)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-222-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-222">influenced(id) = (e,a)</script>.
     </li>
     <li id="axiom9">
-            If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-210-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-210">communicatedBy(id) = (a_2,a_1)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-211-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-211">influenced(id) = (a_2,a_1)</script>.
+        If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-223-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-223">used(id) = (e,a)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-224-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-224">influenced(id) = (e,a)</script>.
     </li>
     <li id="axiom10">
-     If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-212-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-212">started(id) = (a_2,e,a_1)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-213-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-213">influenced(id) = (a_2,e)</script>.
+            If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-225-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-225">communicated(id) = (a_2,a_1)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-226-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-226">influenced(id) = (a_2,a_1)</script>.
     </li>
     <li id="axiom11">
-         If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-214-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-214">ended(id) = (a_2,e,a_1)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-215-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-215">influenced(id) = (a_2,e)</script>.
+     If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-227-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-227">started(id) = (a_2,e,a_1)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-228-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-228">influenced(id) = (a_2,e)</script>.
     </li>
     <li id="axiom12">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-216-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-216">invalidated(id) = (e,a)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-217-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-217">influenced(id) = (e,a)</script>.
+         If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-229-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-229">ended(id) = (a_2,e,a_1)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-230-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-230">influenced(id) = (a_2,e)</script>.
     </li>
     <li id="axiom13">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-218-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">⋅</mo><mi>w</mi><mo stretchy="false">⋅</mo><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-218">derivationPath(id) = e_2 \cdot w \cdot e_1</script> then
-    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-219-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-219">influenced(id) = (e_2,e_1)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-231-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-231">invalidated(id) = (e,a)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-232-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-232">influenced(id) = (e,a)</script>.
     </li>
     <li id="axiom14">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-220-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>T</mi><mi>o</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-220">attributedTo(id) = (e,ag)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-221-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-221">influenced(id) = (e,ag)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-233-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">⋅</mo><mi>w</mi><mo stretchy="false">⋅</mo><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-233">derivationPath(id) = e_2 \cdot w \cdot e_1</script> then
+    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-234-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-234">influenced(id) = (e_2,e_1)</script>.
     </li>
     <li id="axiom15">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-222-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-222">associatedWith(id) = (a,ag,pl)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-223-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-223">influenced(id) = (a,ag)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-235-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>T</mi><mi>o</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-235">attributedTo(id) = (e,ag)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-236-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-236">influenced(id) = (e,ag)</script>.
     </li>
     <li id="axiom16">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-224-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>c</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>o</mi><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-224">actedFor(id) = (ag_2,ag_1)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-225-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-225">influenced(id) = (ag_2,ag_1)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-237-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-237">associatedWith(id) = (a,ag,pl)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-238-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-238">influenced(id) = (a,ag)</script>.
     </li>
     <li id="axiom17">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-226-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><msup><mi>n</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-226">generate(gen) = (e,a) = generated(gen')</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-227-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">=</mo><mi>g</mi><mi>e</mi><msup><mi>n</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-227">gen = gen'</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-239-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>c</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>o</mi><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-239">actedFor(id) = (ag_2,ag_1)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-240-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-240">influenced(id) = (ag_2,ag_1)</script>.
     </li>
     <li id="axiom18">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-228-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>n</mi><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>i</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>n</mi><msup><mi>v</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-228">invalidated(inv) = (e,a) = invalidated(inv')</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-229-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>v</mi><mo stretchy="false">=</mo><mi>i</mi><mi>n</mi><msup><mi>v</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-229">inv=inv'</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-241-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><msup><mi>n</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-241">generated(gen) = (e,a) = generated(gen')</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-242-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">=</mo><mi>g</mi><mi>e</mi><msup><mi>n</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-242">gen = gen'</script>.
     </li>
     <li id="axiom19">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-230-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>s</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-230">started(st) = (a,e_1,a')</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-231-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>s</mi><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-231">started(st') = (a,e_2,a')</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-232-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mo stretchy="false">=</mo><mi>s</mi><msup><mi>t</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-232">st=st'</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-243-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>n</mi><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>i</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>n</mi><msup><mi>v</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-243">invalidated(inv) = (e,a) = invalidated(inv')</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-244-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>v</mi><mo stretchy="false">=</mo><mi>i</mi><mi>n</mi><msup><mi>v</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-244">inv=inv'</script>.
     </li>
     <li id="axiom20">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-233-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-233">ended(end) = (a,e_1,a')</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-234-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msup><mi>d</mi><mo>′</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-234">ended(end') = (a,e_2,a')</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-235-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">=</mo><mi>e</mi><mi>n</mi><msup><mi>d</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-235">end=end'</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-245-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>s</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-245">started(st) = (a,e_1,a')</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-246-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>s</mi><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-246">started(st') = (a,e_2,a')</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-247-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mo stretchy="false">=</mo><mi>s</mi><msup><mi>t</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-247">st=st'</script>.
     </li>
     <li id="axiom21">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-236-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>s</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-236">started(st) = (a,e)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-237-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mo stretchy="false">⪯</mo><mi>e</mi><mi>v</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-237">st \preceq evt</script> for all <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-238-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-238">evt \in events(a)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-248-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-248">ended(end) = (a,e_1,a')</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-249-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msup><mi>d</mi><mo>′</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-249">ended(end') = (a,e_2,a')</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-250-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">=</mo><mi>e</mi><mi>n</mi><msup><mi>d</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-250">end=end'</script>.
     </li>
     <li id="axiom22">
-        If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-239-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-239">ended(end) = (a,e,a') </script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-240-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">⪯</mo><mi>e</mi><mi>n</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-240">evt \preceq end</script> for all <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-241-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-241">evt \in events(a)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-251-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>s</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-251">started(st) = (a,e)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-252-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mo stretchy="false">⪯</mo><mi>e</mi><mi>v</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-252">st \preceq evt</script> for all <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-253-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">−</mo><mi>I</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-253">evt \in
+    events(a) - Invalidations</script>.
     </li>
     <li id="axiom23">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-242-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-242">generated(gen) = (e,a)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-243-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">⪯</mo><mi>e</mi><mi>v</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-243">gen \preceq evt</script> for all <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-244-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-244">evt \in events(e)</script>.
+        If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-254-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-254">ended(end) = (a,e,a') </script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-255-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">⪯</mo><mi>e</mi><mi>n</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-255">evt \preceq end</script> for all
+    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-256-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">−</mo><mi>I</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-256">evt \in events(a) - Invalidations</script>.
     </li>
     <li id="axiom24">
-        If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-245-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>n</mi><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-245">invalidated(inv) = (e,a)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-246-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">⪯</mo><mi>i</mi><mi>n</mi><mi>v</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-246">evt\preceq inv</script> for all <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-247-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-247">evt \in events(e)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-257-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-257">generated(gen) = (e,a)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-258-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">⪯</mo><mi>e</mi><mi>v</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-258">gen \preceq evt</script> for all <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-259-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-259">evt \in events(e)</script>.
     </li>
     <li id="axiom25">
-    For any derivation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-248-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-248">deriv</script>, with path <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-249-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>w</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-249">derivationPath(deriv) = w</script>,
-    if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-250-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">⋅</mo><mi>g</mi><mo stretchy="false">⋅</mo><mi>a</mi><mo stretchy="false">⋅</mo><mi>u</mi><mo stretchy="false">⋅</mo><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-250">e_2 \cdot g \cdot a \cdot u \cdot e_1 </script> is a substring of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-251-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-251">w</script>
-    where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-252-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">∈</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-252">e_1,e_2 \in Entities</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-253-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo stretchy="false">∈</mo><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-253">g \in Generations</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-254-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo stretchy="false">∈</mo><mi>U</mi><mi>s</mi><mi>a</mi><mi>g</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-254">u \in Usages</script>
-    and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-255-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo stretchy="false">∈</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-255">a \in Activities</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-256-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo stretchy="false">⪯</mo><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-256">u \preceq g</script>.
+        If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-260-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>n</mi><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-260">invalidated(inv) = (e,a)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-261-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">⪯</mo><mi>i</mi><mi>n</mi><mi>v</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-261">evt\preceq inv</script> for all
+    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-262-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-262">evt \in events(e)</script>.
     </li>
     <li id="axiom26">
-    For any derivation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-257-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-257">deriv</script>, with path <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-258-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">⋅</mo><mi>w</mi><mo stretchy="false">⋅</mo><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-258">derivationPath(deriv) = e_2
-    \cdot w \cdot e_1</script>, if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-259-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-259">generated(gen_1) = (e_1,a_1)</script> and
-    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-260-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-260">generated(gen_2) = (e_2,a_2)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-261-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mo stretchy="false">≺</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-261">gen_1 \prec gen_2</script>.  
+    For any derivation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-263-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-263">deriv</script>, with path <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-264-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>w</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-264">derivationPath(deriv) = w</script>,
+    if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-265-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">⋅</mo><mi>g</mi><mo stretchy="false">⋅</mo><mi>a</mi><mo stretchy="false">⋅</mo><mi>u</mi><mo stretchy="false">⋅</mo><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-265">e_2 \cdot g \cdot a \cdot u \cdot e_1 </script> is a substring of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-266-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-266">w</script>
+    where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-267-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">∈</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-267">e_1,e_2 \in Entities</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-268-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo stretchy="false">∈</mo><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-268">g \in Generations</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-269-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo stretchy="false">∈</mo><mi>U</mi><mi>s</mi><mi>a</mi><mi>g</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-269">u \in Usages</script>
+    and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-270-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo stretchy="false">∈</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-270">a \in Activities</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-271-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo stretchy="false">⪯</mo><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-271">u \preceq g</script>.
+    </li>
+    <li id="axiom27">
+    For any derivation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-272-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-272">deriv</script>, with path <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-273-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">⋅</mo><mi>w</mi><mo stretchy="false">⋅</mo><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-273">derivationPath(deriv) = e_2
+    \cdot w \cdot e_1</script>, if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-274-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-274">generated(gen_1) = (e_1,a_1)</script> and
+    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-275-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-275">generated(gen_2) = (e_2,a_2)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-276-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mo stretchy="false">≺</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-276">gen_1 \prec gen_2</script>.  
+    </li>
+    <li id="axiom28">
+    If  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-277-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-277">associatedWith(assoc) = (a,ag,pl)</script>  and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-278-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-278">started(start) = (a,e_1,a_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-279-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>n</mi><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-279">invalidated(inv) =
+    (ag,a_2)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-280-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">⪯</mo><mi>i</mi><mi>n</mi><mi>v</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-280">start \preceq inv</script>.
+    </li>
+    <li id="axiom29">
+    If  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-281-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-281">associatedWith(assoc) = (a,ag,pl)</script>  and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-282-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-282">generated(gen) =
+    (ag,a_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-283-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-283">ended(end) = (a,e_2,a_2)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-284-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">⪯</mo><mi>e</mi><mi>n</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-284">gen \preceq end</script>.
+    </li>
+    <li id="axiom30">
+    If  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-285-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-285">associatedWith(assoc) = (a,ag,pl)</script>  and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-286-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-286">started(start) = (a,e_1,a_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-287-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-287">ended(end) =
+    (ag,e_2,a_2)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-288-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">⪯</mo><mi>e</mi><mi>n</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-288">start \preceq end</script>.
+    </li>
+    <li id="axiom31">
+    If  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-289-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-289">associatedWith(assoc) = (a,ag,pl)</script>  and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-290-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-290">started(start) =
+    (ag,e_1,a_1)</script> and <span class="MathJax_Preview"></span><span cla