* fixed typos
authorjcheney@inf.ed.ac.uk
Fri, 05 Apr 2013 18:34:52 +0100
changeset 6065 627aa61fd70d
parent 6064 8b6a3e6e27c8
child 6066 3d236d13a755
* fixed typos
semantics/prov-sem.html
semantics/releases/NOTE-prov-sem-20130430/Overview.html
semantics/releases/NOTE-prov-sem-20130430/prov-sem.pdf
--- a/semantics/prov-sem.html	Fri Apr 05 18:20:28 2013 +0100
+++ b/semantics/prov-sem.html	Fri Apr 05 18:34:52 2013 +0100
@@ -1399,7 +1399,7 @@
 <em>components</em>, highlighted in boxes.
 
 
-
+<section>
 <h3> Things </h3> 
 
 <p><em>Things</em>  is a set of things in the situation being modeled.
@@ -1675,15 +1675,24 @@
 Events$.  We write $e \prec e'$ when $e \preceq e'$ and $e'
 \not\preceq e$ hold.
 </li>
-<li>A function $started : Starts \to Activities \times Entities \times Activities$.
-</li>
-<li>A function $ended : Ends \to Activities \times Entities \times Activities$.
+<li>A function $started : Starts \to Activities \times Entities \times
+Activities$, such that $started(st) = (a,e,a')$ implies $start \in
+events(a) \cap events(e) \cap events(a')$.
 </li>
-<li>A function $used : Usages \to Activities \times Entities$.
+<li>A function $ended : Ends \to Activities \times Entities \times Activities$, such that $ended(end) = (a,e,a')$ implies $end \in
+events(a) \cap events(e) \cap events(a')$.
 </li>
-<li>A function $generated : Generations \to Entities \times Activities$.
+<li>A function $used : Usages \to Activities \times Entities$ such
+that $used(use) = (a,e)$ implies $use \in
+events(a) \cap events(e) \cap events(a')$.
 </li>
-<li>A function $invalidated : Invalidations \to Entities \times Activities$.
+<li>A function $generated : Generations \to Entities \times
+Activities$ such that $generated(gen) = (a,e)$ implies $gen \in
+events(a) \cap events(e)$.
+</li>
+<li>A function $invalidated : Invalidations \to Entities \times
+Activities$ such that $invalidated(inv) = (a,e)$ implies $inv \in
+events(a) \cap events(e)$.
 </li>
 </ol>
 </div>
@@ -1961,10 +1970,11 @@
   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).
+  follow the activity's start event(s) and precede its end
+  event(s).</p>
   </div>
   </section>
-
+  
 <section>
 <h3> Putting it all together </h3>
 
@@ -2013,8 +2023,8 @@
 </p>
 </section>
 
-
 </section>
+
 <section id="semantics">
 <h2> Semantics </h2>
 
--- a/semantics/releases/NOTE-prov-sem-20130430/Overview.html	Fri Apr 05 18:20:28 2013 +0100
+++ b/semantics/releases/NOTE-prov-sem-20130430/Overview.html	Fri Apr 05 18:34:52 2013 +0100
@@ -908,7 +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="#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>
+</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><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>
 
 
 
@@ -1281,8 +1281,8 @@
 <em>components</em>, highlighted in boxes.
 
 
-
-</p><h2 id="things-1"> Things </h2> 
+</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 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
@@ -1328,7 +1328,7 @@
 </section>
 
 <section id="objects-1">
-<!--OddPage--><h2><span class="secno">4. </span> Objects </h2>
+<h3><span class="secno">3.2 </span> Objects </h3>
 
 
 <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
@@ -1381,7 +1381,7 @@
   </div>
   
 <section id="entities-1">
-<h3><span class="secno">4.1 </span> Entities </h3>
+<h4><span class="secno">3.2.1 </span> Entities </h4>
 
 <p>An <em>entity</em> is a kind of object that fixes some aspects of a
   thing. We assume:</p>
@@ -1455,7 +1455,7 @@
   </div>
   
 <section id="plans-1">  
-<h4><span class="secno">4.1.1 </span> Plans </h4>
+<h5><span class="secno">3.2.1.1 </span> Plans </h5>
 <p>We identify a specific subset of the entities called
   <em>plans</em>:</p>
 <div class="component" id="plans" data-count="4" data-title="Component 4 (plans)"><div class="ruleTitle"><a class="internalDFN" href="#plans">Component 4 (plans)</a></div>
@@ -1464,7 +1464,7 @@
 </section>
 
 <section id="collections-1">
-  <h4><span class="secno">4.1.2 </span>Collections</h4>
+  <h5><span class="secno">3.2.1.2 </span>Collections</h5>
   <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>
@@ -1478,7 +1478,7 @@
   </section>
 
     <section id="activities-1">
-<h3><span class="secno">4.2 </span> Activities </h3>
+<h4><span class="secno">3.2.2 </span> Activities </h4>
 
 
 <p>An <em>activity</em> is an object that encompasses a set of events.  We introduce:
@@ -1493,7 +1493,7 @@
 </div></section>
   
   <section id="agents-1">
-<h3><span class="secno">4.3 </span> Agents </h3>
+<h4><span class="secno">3.2.3 </span> Agents </h4>
 
 <p>An agent is an object that can act, by controlling, starting,
   ending, or participating in activities.  An agent is something that
@@ -1513,7 +1513,7 @@
 
 
 <section id="influences-1">
-<h3><span class="secno">4.4 </span> Influences </h3>
+<h4><span class="secno">3.2.4 </span> Influences </h4>
 
 <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
@@ -1541,7 +1541,7 @@
 
 
 <section id="events-1">
-<h4><span class="secno">4.4.1 </span> Events </h4>
+<h5><span class="secno">3.2.4.1 </span> Events </h5>
 
 <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
@@ -1557,112 +1557,121 @@
 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-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-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>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>, such that <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>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><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-159">started(st) = (a,e,a')</script> implies <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>s</mi><mi>t</mi><mi>a</mi><mi>r</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>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><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-160">start \in
+events(a) \cap events(e) \cap events(a')</script>.
 </li>
-<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>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>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-161">ended : Ends \to Activities \times Entities \times Activities</script>, such that <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>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-162">ended(end) = (a,e,a')</script> implies <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>e</mi><mi>n</mi><mi>d</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>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><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-163">end \in
+events(a) \cap events(e) \cap events(a')</script>.
 </li>
-<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>A 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>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-164">used : Usages \to Activities \times Entities</script> such
+that <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>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>u</mi><mi>s</mi><mi>e</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-165">used(use) = (a,e)</script> implies <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>u</mi><mi>s</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>a</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>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><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-166">use \in
+events(a) \cap events(e) \cap events(a')</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>A function <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>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-167">generated : Generations \to Entities \times
+Activities</script> such that <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>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><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-168">generated(gen) = (a,e)</script> implies <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>g</mi><mi>e</mi><mi>n</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>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-169">gen \in
+events(a) \cap events(e)</script>.
+</li>
+<li>A function <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>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-170">invalidated : Invalidations \to Entities \times
+Activities</script> such that <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>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><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-171">invalidated(inv) = (a,e)</script> implies <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>i</mi><mi>n</mi><mi>v</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>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-172">inv \in
+events(a) \cap events(e)</script>.
 </li>
 </ol>
 </div>
 </section>
 <section id="associations-1">
 
-<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
+<h5><span class="secno">3.2.4.2 </span> Associations </h5>
+
+<p>An <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>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-173">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-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>A set <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>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-174">Associations \subseteq Influences</script> with associated
+  function <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>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-175">associatedWith : Associations \to  Agents \times Activities \times Plans_\bot</script>.
 </p>
   </div>
   </section>
 <section id="attributions-1">
 
-<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:
+<h5><span class="secno">3.2.4.3 </span> Attributions </h5>
+
+<p>An <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>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-176">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-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>A set <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>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-177">Attributions \subseteq Influences</script> with associated
+  function <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>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-178">attributedTo : Attributions \to Entities \times Agents</script>.
 </p>
   </div>
   
 </section>
   <section id="communications-1">
-  <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
+  <h5><span class="secno">3.2.4.4 </span>Communications</h5>
+  <p>A <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"><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-179">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-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>A set <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"><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-180">Communications \subseteq Influences</script> with associated
+  function <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>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-181">communicated : Communications \to Activities \times Activities</script>.
 </p>
   </div>
   
 
 </section>
   <section id="delegations-1">
-  <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
+  <h5><span class="secno">3.2.4.5 </span>Delegations</h5>
+<p>A <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>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-182">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-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>A set <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>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-183">Delegations \subseteq Influences</script> and associated function
+  <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><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-184">actedFor : Delegations \to Agents \times Agents \times Activities</script>
 </p>
   </div>
   
 </section>
   <section id="derivations-1">
   
-  <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
+  <h5><span class="secno">3.2.4.6 </span> Derivations </h5>
+
+<p>A <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>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-185">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-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
+<span class="MathJax_Preview"></span><div 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" 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-186">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-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>where the <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>e</mi><mi>n</mi><msub><mi>t</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-187">ent_i</script> are entities, <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>a</mi><mi>c</mi><msub><mi>t</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-188">act_i</script> are activities, <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"><msub><mi>g</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-189">g_i</script> are generations, and <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>u</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-190">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-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
+<span class="MathJax_Preview"></span><div 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" 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-191">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-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>for each substring <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>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-192">ent\cdot g \cdot act</script> we have <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>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-193">generated(g) = (ent,act)</script>, and
 </li>
-<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>for each substring <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>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-194">act \cdot u \cdot ent</script> we have <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><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-195">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-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
+<p>  A set <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>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-196">Derivations \subseteq Influences</script> with an associated
+  function <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>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-197">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-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
+  The <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>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-198">derivationPath</script> function links each <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>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-199"> 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-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>.
+  instance, there may be multiple derivation paths linking <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"><msub><mi>e</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-200">e_2</script> to
+  <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"><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-201">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-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>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-202">d \in Derivations</script>.
   </p>
 
   <p>A derivation path implies the existence of at least one chained generation
@@ -1675,7 +1684,7 @@
   </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-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
+  steps is to reflect that <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>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-203">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
@@ -1686,7 +1695,7 @@
 </section>
 
   <section id="additional-axioms">
-  <!--OddPage--><h2><span class="secno">5. </span>Additional axioms</h2>
+  <h3><span class="secno">3.3 </span>Additional axioms</h3>
 
   <p> Above we have stated some properties of the components.  We
   impose some additional properties that relate several components, as
@@ -1696,135 +1705,135 @@
     
   <ol>
     <li id="axiom1">
-    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>.
+    If <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><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(g) = (e,a_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>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-205">used(u) = (a_2,e)</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>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-206">c \in Communications</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>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-207">communicated(c) = (a_2,a_1)</script>.
     </li>
     <li id="axiom2">
-    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>.
+    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>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-208">e \in Entities</script> then there exist <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>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-209">gen,inv,a,a'</script> such that
+    <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>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-210">generated(gen) = (e,a)</script> and <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>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-211">invalidated(inv) = (e,a')</script>.
     </li>
     <li id="axiom3">
-    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>.
+    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>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-212">started(start) = (a_2,e,a_1)</script> then there exists <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> such
+    that <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>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-214">generated(gen) = (e,a_1)</script>.
     </li>
     <li id="axiom4">
-    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>.
+    If <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>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-215">ended(end) = (a_2,e,a_1)</script> then there exists <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>g</mi><mi>e</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-216">gen</script> such
+    that <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>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-217">generated(gen) = (e,a_1)</script>.
     </li>
     <li id="axiom5">
-    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>.
+    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><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-218">d \in Derivations</script> and <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>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-219">prov:Revision \in
+    value(d,prov:type)</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>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-220">derivationPath(deriv) = e_2 \cdot w \cdot
+    e_1</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>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-221">thingOf(e_1) = thingOf(e_2)</script>.
     </li>
     <li id="axiom6">
-    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>.
+    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>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-222">attributedTo(att) = (e,ag)</script> then there exist <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>g</mi><mi>e</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-223">gen</script> and <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>s</mi><mi>s</mi><mi>o</mi><mi>c</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-224">assoc</script>
+    such that <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>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-225">generated(gen) = (e,a)</script> and <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>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-226">associatedWith(assoc) = (a,ag)</script>.
     </li>
     <li id="axiom7">
-    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>.
+    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>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-227">actedFor(deleg) = (ag_2,ag_1,act)</script> then there exist
+    <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>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-228">assoc_1,assoc_2,pl_1,pl_2</script> such that <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>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-229">associatedWith(assoc_1) = (ag_1,act,pl_1)</script>
+    and <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>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-230">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-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>.
+    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>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-231">generated(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="axiom9">
-        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>.
+        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>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-233">used(id) = (e,a)</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><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-234">influenced(id) = (e,a)</script>.
     </li>
     <li id="axiom10">
-            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>.
+            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>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-235">communicated(id) = (a_2,a_1)</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><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-236">influenced(id) = (a_2,a_1)</script>.
     </li>
     <li id="axiom11">
-     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>.
+     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>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-237">started(id) = (a_2,e,a_1)</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><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-238">influenced(id) = (a_2,e)</script>.
     </li>
     <li id="axiom12">
-         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>.
+         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>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-239">ended(id) = (a_2,e,a_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><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-240">influenced(id) = (a_2,e)</script>.
     </li>
     <li id="axiom13">
-    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>.
+    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>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-241">invalidated(id) = (e,a)</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>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-242">influenced(id) = (e,a)</script>.
     </li>
     <li id="axiom14">
-    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>.
+    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>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-243">derivationPath(id) = e_2 \cdot w \cdot e_1</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>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-244">influenced(id) = (e_2,e_1)</script>.
     </li>
     <li id="axiom15">
-    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>.
+    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>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-245">attributedTo(id) = (e,ag)</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>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-246">influenced(id) = (e,ag)</script>.
     </li>
     <li id="axiom16">
-    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>.
+    If <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>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-247">associatedWith(id) = (a,ag,pl)</script> then <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>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-248">influenced(id) = (a,ag)</script>.
     </li>
     <li id="axiom17">
-    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>.
+    If <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>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-249">actedFor(id) = (ag_2,ag_1)</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>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-250">influenced(id) = (ag_2,ag_1)</script>.
     </li>
     <li id="axiom18">
-    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>.
+    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>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-251">generated(gen) = (e,a) = generated(gen')</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>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-252">gen = gen'</script>.
     </li>
     <li id="axiom19">
-    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>.
+    If <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>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-253">invalidated(inv) = (e,a) = invalidated(inv')</script> then <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>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-254">inv=inv'</script>.
     </li>
     <li id="axiom20">
-    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>.
+    If <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>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-255">started(st) = (a,e_1,a')</script> and <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>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-256">started(st') = (a,e_2,a')</script> then <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>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-257">st=st'</script>.
     </li>
     <li id="axiom21">
-    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>.
+    If <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>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-258">ended(end) = (a,e_1,a')</script> and <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>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-259">ended(end') = (a,e_2,a')</script> then <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>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-260">end=end'</script>.
     </li>
     <li id="axiom22">
-    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
+    If <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>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-261">started(st) = (a,e)</script> then <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>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-262">st \preceq evt</script> for all <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>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-263">evt \in
     events(a) - Invalidations</script>.
     </li>
     <li id="axiom23">
-        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>.
+        If <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>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-264">ended(end) = (a,e,a') </script> then <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"><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-265">evt \preceq end</script> for all
+    <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>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-266">evt \in events(a) - Invalidations</script>.
     </li>
     <li id="axiom24">
-    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>.
+    If <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"><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-267">generated(gen) = (e,a)</script> then <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><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-268">gen \preceq evt</script> for all <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>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-269">evt \in events(e)</script>.
     </li>
     <li id="axiom25">
-        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>.
+        If <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>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-270">invalidated(inv) = (e,a)</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>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-271">evt\preceq inv</script> for all
+    <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>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-272">evt \in events(e)</script>.
     </li>
     <li id="axiom26">
-    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>.
+    For any derivation <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></math></span></span></span><script type="math/tex" id="MathJax-Element-273">deriv</script>, with path <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>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-274">derivationPath(deriv) = w</script>,
+    if <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"><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-275">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-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>w</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-276">w</script>
+    where <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"><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-277">e_1,e_2 \in Entities</script>, <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>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-278">g \in Generations</script>, <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>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-279">u \in Usages</script>
+    and <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>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-280">a \in Activities</script> then <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>u</mi><mo stretchy="false">⪯</mo><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-281">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>.  
+    For any derivation <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>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-282">deriv</script>, with path <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>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-283">derivationPath(deriv) = e_2
+    \cdot w \cdot e_1</script>, if <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><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-284">generated(gen_1) = (e_1,a_1)</script> and
+    <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>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-285">generated(gen_2) = (e_2,a_2)</script> then <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>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-286">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>.
+    If  <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>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-287">associatedWith(assoc) = (a,ag,pl)</script>  and <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><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-288">started(start) = (a,e_1,a_1)</script> and <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>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-289">invalidated(inv) =
+    (ag,a_2)</script> then <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><mo stretchy="false">⪯</mo><mi>i</mi><mi>n</mi><mi>v</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-290">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>.
+    If  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-291-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-291">associatedWith(assoc) = (a,ag,pl)</script>  and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-292-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-292">generated(gen) =
+    (ag,a_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-293-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-293">ended(end) = (a,e_2,a_2)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-294-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-294">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>.
+    If  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-295-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-295">associatedWith(assoc) = (a,ag,pl)</script>  and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-296-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-296">started(start) = (a,e_1,a_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-297-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-297">ended(end) =
+    (ag,e_2,a_2)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-298-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-298">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 class="MathJax_MathML" id="MathJax-Element-291-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-291">ended(end) = (a,e_2,a_2)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-292-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-292">start \preceq end</script>.
+    If  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-299-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-299">associatedWith(assoc) = (a,ag,pl)</script>  and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-300-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-300">started(start) =
+    (ag,e_1,a_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-301-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-301">ended(end) = (a,e_2,a_2)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-302-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-302">start \preceq end</script>.
     </li>
        <li id="axiom32">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-293-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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-293">attributedTo(attrib) = (e,ag)</script>  and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-294-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>a</mi><msub><mi>g</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-294">generated(gen_1) =
-    (ag_1,a_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-295-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>e</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-295">generated(gen_2) = (e,a_2)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-296-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-296">gen_1 \preceq gen_2</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-303-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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-303">attributedTo(attrib) = (e,ag)</script>  and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-304-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>a</mi><msub><mi>g</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-304">generated(gen_1) =
+    (ag_1,a_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-305-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>e</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-305">generated(gen_2) = (e,a_2)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-306-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-306">gen_1 \preceq gen_2</script>.
     </li>
        <li id="axiom33">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-297-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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-297">attributedTo(attrib) = (e,ag)</script>  and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-298-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>g</mi><mn>1</mn></msub><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-298">started(start) =
-    (ag_1,e_1,a_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-299-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-299">generated(gen) = (e,a_2)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-300-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi><mi>e</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-300">start \preceq gen</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-307-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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-307">attributedTo(attrib) = (e,ag)</script>  and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-308-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>g</mi><mn>1</mn></msub><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-308">started(start) =
+    (ag_1,e_1,a_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-309-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-309">generated(gen) = (e,a_2)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-310-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi><mi>e</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-310">start \preceq gen</script>.
     </li>
        <li id="axiom34">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-301-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-301">actedFor(deleg) = (ag_2,ag_1,a)</script>  and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-302-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>g</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-302">generated(gen) =
-    (ag_1,a_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-303-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>g</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-303">invalidated(inv) = (ag_2,a_2)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-304-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-304">gen \preceq inv</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-311-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-311">actedFor(deleg) = (ag_2,ag_1,a)</script>  and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-312-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>g</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-312">generated(gen) =
+    (ag_1,a_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-313-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>g</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-313">invalidated(inv) = (ag_2,a_2)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-314-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-314">gen \preceq inv</script>.
     </li>
        <li id="axiom35">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-305-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-305">actedFor(deleg) = (ag_2,ag_1,a)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-306-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>g</mi><mn>1</mn></msub><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-306">started(start) =
-    (ag_1,e_1,a_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-307-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>g</mi><mn>2</mn></msub><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-307">ended(end) = (ag_2,e_2,a_2)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-308-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-308">start \preceq
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-315-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-315">actedFor(deleg) = (ag_2,ag_1,a)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-316-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>g</mi><mn>1</mn></msub><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-316">started(start) =
+    (ag_1,e_1,a_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-317-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>g</mi><mn>2</mn></msub><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-317">ended(end) = (ag_2,e_2,a_2)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-318-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-318">start \preceq
     end</script>.
     </li>
     </ol>
@@ -1842,17 +1851,18 @@
   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).
-  </p></div>
+  follow the activity's start event(s) and precede its end
+  event(s).</p>
+  </div>
   </section>
-
+  
 <section id="putting-it-all-together">
-<!--OddPage--><h2><span class="secno">6. </span> Putting it all together </h2>
-
-<p>A <em>PROV structure</em> <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-309-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-309">W</script> is a collection of sets, functions, and relations containing all of the above
+<h3><span class="secno">3.4 </span> Putting it all together </h3>
+
+<p>A <em>PROV structure</em> <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-319-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-319">W</script> 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 <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-310-Frame" style="font-size: 100%; "><span class="MathJax_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>W</mi><mn>1</mn></msub><mo>.</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-310">W_1.Objects</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-311-Frame" style="font-size: 100%; "><span class="MathJax_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>W</mi><mn>1</mn></msub><mo>.</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-311">W_1.Things</script>,
+more than one structure then we may write <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-320-Frame" style="font-size: 100%; "><span class="MathJax_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>W</mi><mn>1</mn></msub><mo>.</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-320">W_1.Objects</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-321-Frame" style="font-size: 100%; "><span class="MathJax_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>W</mi><mn>1</mn></msub><mo>.</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-321">W_1.Things</script>,
 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>
@@ -1861,9 +1871,9 @@
 <p>
 Some features of PROV structures are relatively obvious or routine,
 corresponding directly to features of PROV and associated inferences.
-For example, the functions <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-312-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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>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>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>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-312">used, generated, invalidated, started,
+For example, the functions <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-322-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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>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>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>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-322">used, generated, invalidated, started,
 ended</script> mapping events to their associated entities or activities, and
-<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-313-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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>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>c</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>o</mi><mi>r</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-313">communicated, associatedWith, attributedTo, actedFor</script> associating
+<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-323-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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>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>c</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>o</mi><mi>r</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-323">communicated, associatedWith, attributedTo, actedFor</script> associating
 other types of influences with appropriate data. 
 </p>
   <p>
@@ -1873,118 +1883,118 @@
 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 <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-314-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-314">Things</script>, which correspond to changing, real-world
-things, as opposed to <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-315-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-315">Entities</script>, which correspond to limited views or
-perspectives on <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-316-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-316">Things</script>, with some fixed aspects.  The semantic
-structures of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-317-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-317">Things</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-318-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-318">Entities</script> provides a foundation for the
-<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-319-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-319">alternateOf</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-320-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-320">specializationOf</script> relations.
+is the use of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-324-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-324">Things</script>, which correspond to changing, real-world
+things, as opposed to <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-325-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-325">Entities</script>, which correspond to limited views or
+perspectives on <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-326-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-326">Things</script>, with some fixed aspects.  The semantic
+structures of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-327-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-327">Things</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-328-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-328">Entities</script> provides a foundation for the
+<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-329-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-329">alternateOf</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-330-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-330">specializationOf</script> relations.
 </p>
   
 
 
 </section>
 <section id="interpretations">
-<!--OddPage--><h2><span class="secno">7. </span> Interpretations </h2>
+<h3><span class="secno">3.5 </span> Interpretations </h3>
 
 <p>We need to link identifiers to the objects they denote.  We do this using a function which we shall call an <em>interpretation</em>.
- An interpretation is a function  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-321-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">:</mo><mi>I</mi><mi>d</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>f</mi><mi>i</mi><mi>e</mi><mi>r</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-321">\rho : Identifiers \to Objects</script> describing
+ An interpretation is a function  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-331-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">:</mo><mi>I</mi><mi>d</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>f</mi><mi>i</mi><mi>e</mi><mi>r</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-331">\rho : Identifiers \to Objects</script> describing
 which object is the target of each identifier. The mapping from
  identifiers to objects may <b>not</b> change over time; only
- <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-322-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-322">Objects</script> can be denoted by <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-323-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>f</mi><mi>i</mi><mi>e</mi><mi>r</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-323">Identifiers</script>.
+ <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-332-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-332">Objects</script> can be denoted by <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-333-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>f</mi><mi>i</mi><mi>e</mi><mi>r</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-333">Identifiers</script>.
 </p>
 </section>
 
-
+</section>
 
 <section id="semantics">
-<!--OddPage--><h2><span class="secno">8. </span> Semantics </h2>
-
-<p>In what follows, let <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-324-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-324">W</script> be a fixed structure with the associated sets and relations discussed in the previous section, and let <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-325-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-325">\rho</script> be an interpretation of identifiers as objects in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-326-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-326">W</script>.
+<!--OddPage--><h2><span class="secno">4. </span> Semantics </h2>
+
+<p>In what follows, let <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-334-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-334">W</script> be a fixed structure with the associated sets and relations discussed in the previous section, and let <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-335-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-335">\rho</script> be an interpretation of identifiers as objects in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-336-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-336">W</script>.
 The annotations [WF] refer to well-formedness constraints that correspond to typing constraints.
 </p>
 
 <section id="satisfaction">
-<h3><span class="secno">8.1 </span> Satisfaction </h3>
-
-<p>Consider a formula <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-327-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-327">\phi</script>, a structure <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-328-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-328">W</script> and an interpretation
- <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-329-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-329">\rho</script>.
-We define notation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-330-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-330">W,\rho \models \phi</script> which means that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-331-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-331">\phi</script> is
- satisfied in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-332-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-332">W,\rho</script>. For atomic formulas, the definition of the
+<h3><span class="secno">4.1 </span> Satisfaction </h3>
+
+<p>Consider a formula <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-337-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-337">\phi</script>, a structure <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-338-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-338">W</script> and an interpretation
+ <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-339-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-339">\rho</script>.
+We define notation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-340-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-340">W,\rho \models \phi</script> which means that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-341-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-341">\phi</script> is
+ satisfied in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-342-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-342">W,\rho</script>. For atomic formulas, the definition of the
  satisfaction relation is given in the next few subsections.  We give
  the standard definition of the semantics of the other formulas:
 </p>
 
 <div class="semantics" id="first-order-logic-semantics" data-count="16" data-title="Semantics 16 (first-order-logic-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#first-order-logic-semantics">Semantics 16 (first-order-logic-semantics)</a></div>
 <ol>
-  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-333-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi>T</mi><mi>r</mi><mi>u</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-333">W,\rho \models True</script> always holds.</li>
-  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-334-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-334">W,\rho \models False</script> never holds.</li>
-  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-335-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi>x</mi><mo stretchy="false">=</mo><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-335">W,\rho \models x = y</script> holds if and only if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-336-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-336">\rho(x) =
+  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-343-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi>T</mi><mi>r</mi><mi>u</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-343">W,\rho \models True</script> always holds.</li>
+  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-344-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-344">W,\rho \models False</script> never holds.</li>
+  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-345-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi>x</mi><mo stretchy="false">=</mo><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-345">W,\rho \models x = y</script> holds if and only if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-346-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-346">\rho(x) =
     \rho(y)</script>.</li>
-  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-337-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="normal">¬</mi><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-337">W,\rho \models \neg \phi</script> holds if and only if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-338-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-338">W,\rho \models
+  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-347-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="normal">¬</mi><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-347">W,\rho \models \neg \phi</script> holds if and only if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-348-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-348">W,\rho \models
   \phi</script> does not hold.</li>
-  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-339-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi><mo stretchy="false">∧</mo><mi mathvariant="italic">ψ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-339">W,\rho \models \phi \wedge \psi</script> holds if and only if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-340-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-340">W,\rho \models
-  \phi</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-341-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ψ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-341">W,\rho \models \psi</script>.</li>
-  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-342-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi><mo stretchy="false">∨</mo><mi mathvariant="italic">ψ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-342">W,\rho \models \phi \vee \psi</script> holds if either <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-343-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-343">W,\rho \models \phi</script>
-  or <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-344-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ψ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-344">W,\rho \models \psi</script>.</li>
-  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-345-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi><mo stretchy="false">⇒</mo><mi mathvariant="italic">ψ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-345">W,\rho \models \phi \Rightarrow \psi</script> holds if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-346-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-346">W,\rho \models \phi</script>
-  implies <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-347-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ψ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-347">W,\rho \models \psi</script>.</li>
-  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-348-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="normal">∃</mi><mi>x</mi><mo>.</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-348">W,\rho \models \exists x. \phi</script> holds if there exists some <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-349-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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-349">obj \in
-  Objects</script> such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-350-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">:=</mo><mi>o</mi><mi>b</mi><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-350">W,\rho[x:=obj] \models \phi</script>.</li>
-  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-351-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="normal">∀</mi><mi>x</mi><mo>.</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-351">W,\rho \models \forall x. \phi</script> holds if there for every <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-352-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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-352">obj \in
-  Objects</script> we have <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-353-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">:=</mo><mi>o</mi><mi>b</mi><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-353">W,\rho[x:=obj] \models \phi</script>.</li>
+  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-349-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi><mo stretchy="false">∧</mo><mi mathvariant="italic">ψ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-349">W,\rho \models \phi \wedge \psi</script> holds if and only if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-350-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-350">W,\rho \models
+  \phi</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-351-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ψ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-351">W,\rho \models \psi</script>.</li>
+  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-352-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi><mo stretchy="false">∨</mo><mi mathvariant="italic">ψ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-352">W,\rho \models \phi \vee \psi</script> holds if either <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-353-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-353">W,\rho \models \phi</script>
+  or <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-354-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ψ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-354">W,\rho \models \psi</script>.</li>
+  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-355-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi><mo stretchy="false">⇒</mo><mi mathvariant="italic">ψ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-355">W,\rho \models \phi \Rightarrow \psi</script> holds if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-356-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-356">W,\rho \models \phi</script>
+  implies <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-357-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="italic">ψ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-357">W,\rho \models \psi</script>.</li>
+  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-358-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="normal">∃</mi><mi>x</mi><mo>.</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-358">W,\rho \models \exists x. \phi</script> holds if there exists some <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-359-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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-359">obj \in
+  Objects</script> such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-360-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">:=</mo><mi>o</mi><mi>b</mi><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-360">W,\rho[x:=obj] \models \phi</script>.</li>
+  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-361-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi mathvariant="normal">∀</mi><mi>x</mi><mo>.</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-361">W,\rho \models \forall x. \phi</script> holds if there for every <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-362-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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-362">obj \in
+  Objects</script> we have <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-363-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">[</mo><mi>x</mi><mo stretchy="false">:=</mo><mi>o</mi><mi>b</mi><mi>j</mi><mo stretchy="false">]</mo><mo stretchy="false">⊨</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-363">W,\rho[x:=obj] \models \phi</script>.</li>
   </ol></div>
 
 <div class="remark">
   <p>In the semantics above, note that the domain of quantification is
-  the set of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-354-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-354">Objects</script>; that is, quantifiers range over entities,
+  the set of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-364-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-364">Objects</script>; that is, quantifiers range over entities,
   activities, agents, or influences (which are in turn further
-  subdivided into types of influences).  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-355-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-355">Things</script> and relations
+  subdivided into types of influences).  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-365-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-365">Things</script> and relations
   cannot be referenced directly by identifiers.  
 </p>
   </div>
 
   <div class="remark">
-    <p>A PROV instance <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-356-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-356">I</script> consists of a set of statements, each of
+    <p>A PROV instance <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-366-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-366">I</script> consists of a set of statements, each of
     which can be translated to an atomic formula following the
     definitional rules in PROV-CONSTRAINTS, possibly by introducing
     fresh existential variables.  Thus, we can view an
-    instance <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-357-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-357">I</script> as a set of atomic formulas <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-358-Frame" style="font-size: 100%; "><span class="MathJax_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><msub><mi mathvariant="italic">ϕ</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi mathvariant="italic">ϕ</mi><mi>n</mi></msub><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-358">\{\phi_1,\ldots,\phi_n\}</script>, or equivalently a
-    single formula <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-359-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>x</mi><mi>k</mi></msub><mo>.</mo><mtext>&nbsp;</mtext><msub><mi mathvariant="italic">ϕ</mi><mn>1</mn></msub><mo stretchy="false">∧</mo><mo stretchy="false">⋯</mo><mo stretchy="false">∧</mo><msub><mi mathvariant="italic">ϕ</mi><mi>n</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-359">\exists x_1,\ldots,x_k.~\phi_1 \wedge \cdots
-    \wedge \phi_n</script>, where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-360-Frame" style="font-size: 100%; "><span class="MathJax_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>x</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>x</mi><mi>k</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-360">x_1,\ldots,x_k</script> are the existential
-    variables of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-361-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-361">I</script>.
+    instance <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-367-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-367">I</script> as a set of atomic formulas <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-368-Frame" style="font-size: 100%; "><span class="MathJax_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><msub><mi mathvariant="italic">ϕ</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi mathvariant="italic">ϕ</mi><mi>n</mi></msub><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-368">\{\phi_1,\ldots,\phi_n\}</script>, or equivalently a
+    single formula <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-369-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>x</mi><mi>k</mi></msub><mo>.</mo><mtext>&nbsp;</mtext><msub><mi mathvariant="italic">ϕ</mi><mn>1</mn></msub><mo stretchy="false">∧</mo><mo stretchy="false">⋯</mo><mo stretchy="false">∧</mo><msub><mi mathvariant="italic">ϕ</mi><mi>n</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-369">\exists x_1,\ldots,x_k.~\phi_1 \wedge \cdots
+    \wedge \phi_n</script>, where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-370-Frame" style="font-size: 100%; "><span class="MathJax_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>x</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>x</mi><mi>k</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-370">x_1,\ldots,x_k</script> are the existential
+    variables of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-371-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-371">I</script>.
     </p>
     </div>
 
 </section>
     <section id="attribute-matching">
     
-<h3><span class="secno">8.2 </span> Attribute matching </h3>
-
-
-<p>We say that an object <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-362-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-362">obj</script> matches attributes <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-363-Frame" style="font-size: 100%; "><span class="MathJax_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>a</mi><mi>t</mi><mi>t</mi><msub><mi>r</mi><mn>1</mn></msub><mo stretchy="false">=</mo><mi>v</mi><mi>a</mi><msub><mi>l</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-363">[attr_1=val_1,...]</script> in structure <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-364-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-364">W</script> provided:
-for each attribute <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-365-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>r</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-365">attr_i</script>, we have <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-366-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>l</mi><mi>i</mi></msub><mo stretchy="false">∈</mo><mi>W</mi><mo>.</mo><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><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-366">val_i \in W.value(obj,attr_i)</script>.
-This is sometimes abbreviated as: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-367-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>o</mi><mi>b</mi><mi>j</mi><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-367">match(W,obj,attrs)</script>.
+<h3><span class="secno">4.2 </span> Attribute matching </h3>
+
+
+<p>We say that an object <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-372-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-372">obj</script> matches attributes <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-373-Frame" style="font-size: 100%; "><span class="MathJax_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>a</mi><mi>t</mi><mi>t</mi><msub><mi>r</mi><mn>1</mn></msub><mo stretchy="false">=</mo><mi>v</mi><mi>a</mi><msub><mi>l</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-373">[attr_1=val_1,...]</script> in structure <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-374-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-374">W</script> provided:
+for each attribute <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-375-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>r</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-375">attr_i</script>, we have <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-376-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>l</mi><mi>i</mi></msub><mo stretchy="false">∈</mo><mi>W</mi><mo>.</mo><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><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><msub><mi>r</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-376">val_i \in W.value(obj,attr_i)</script>.
+This is sometimes abbreviated as: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-377-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>o</mi><mi>b</mi><mi>j</mi><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-377">match(W,obj,attrs)</script>.
 </p>
     
 </section>
 
 <section id="semantics-of-element-formulas">
-<h3><span class="secno">8.3 </span> Semantics of Element Formulas </h3>
+<h3><span class="secno">4.3 </span> Semantics of Element Formulas </h3>
 
 <section id="entity">
 
-<h4><span class="secno">8.3.1 </span> Entity </h4>
-
-<p>An entity formula is of the form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-368-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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></math></span></span></span><script type="math/tex" id="MathJax-Element-368">entity(id,attrs)</script> where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-369-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-369">id</script> denotes an entity.
+<h4><span class="secno">4.3.1 </span> Entity </h4>
+
+<p>An entity formula is of the form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-378-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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></math></span></span></span><script type="math/tex" id="MathJax-Element-378">entity(id,attrs)</script> where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-379-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-379">id</script> denotes an entity.
 </p>
-<p>Entity formulas <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-370-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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></math></span></span></span><script type="math/tex" id="MathJax-Element-370">entity(id,attrs)</script> can be interpreted as follows:
+<p>Entity formulas <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-380-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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></math></span></span></span><script type="math/tex" id="MathJax-Element-380">entity(id,attrs)</script> can be interpreted as follows:
 </p>
 <div class="semantics" id="entity-semantics" data-count="17" data-title="Semantics 17 (entity-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#entity-semantics">Semantics 17 (entity-semantics)</a></div>
- <p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-371-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-371">W,\rho \models entity(id,attrs)</script> holds if and only if:</p>
+ <p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-381-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-381">W,\rho \models entity(id,attrs)</script> holds if and only if:</p>
 <ol>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-372-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-372">id</script> denotes an entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-373-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</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></math></span></span></span><script type="math/tex" id="MathJax-Element-373">ent = \rho(id) \in Entities</script>
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-382-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-382">id</script> denotes an entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-383-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</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></math></span></span></span><script type="math/tex" id="MathJax-Element-383">ent = \rho(id) \in Entities</script>
 </li>
-<li>the attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-374-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>e</mi><mi>n</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></math></span></span></span><script type="math/tex" id="MathJax-Element-374">match(W,ent, attrs)</script>.
+<li>the attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-384-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>e</mi><mi>n</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></math></span></span></span><script type="math/tex" id="MathJax-Element-384">match(W,ent, attrs)</script>.
 </li>
 </ol>
 
@@ -1992,7 +2002,7 @@
 </div>
 <div class="remark">
 <p>Not all of the attributes of an entity object are
-  required to be present in an entity formula about that object.  For example, the following formulas all hold if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-375-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-375">x</script> denotes an entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-376-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-376">e</script> such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-377-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo fence="false" stretchy="false">{</mo><mn>4</mn><mo stretchy="false">,</mo><mn>5</mn><mo fence="false" 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>e</mi><mo stretchy="false">,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><mn>6</mn><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-377">value(e,a) = \{4,5\}, value(e,b) = \{6\}</script> hold:
+  required to be present in an entity formula about that object.  For example, the following formulas all hold if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-385-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-385">x</script> denotes an entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-386-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-386">e</script> such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-387-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo fence="false" stretchy="false">{</mo><mn>4</mn><mo stretchy="false">,</mo><mn>5</mn><mo fence="false" 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>e</mi><mo stretchy="false">,</mo><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><mn>6</mn><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-387">value(e,a) = \{4,5\}, value(e,b) = \{6\}</script> hold:
 </p><pre> entity(x,[])
  entity(x,[a=5])
  entity(x,[a=4,a=5])
@@ -2008,44 +2018,44 @@
   </section>
 <section id="activity">
 
-<h4><span class="secno">8.3.2 </span> Activity </h4>
-
-<p>An activity formula  is of the form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-378-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-378">activity(id,st,et,attrs)</script>
-where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-379-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-379">id</script> is a identifier referring to the activity, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-380-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-380">st</script> is a start
-time and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-381-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-381">et</script> is an end time, and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-382-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-382">attrs</script> are the attributes of
-activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-383-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-383">id</script>.
+<h4><span class="secno">4.3.2 </span> Activity </h4>
+
+<p>An activity formula  is of the form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-388-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-388">activity(id,st,et,attrs)</script>
+where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-389-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-389">id</script> is a identifier referring to the activity, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-390-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-390">st</script> is a start
+time and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-391-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-391">et</script> is an end time, and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-392-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-392">attrs</script> are the attributes of
+activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-393-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-393">id</script>.
 </p>
 <div class="semantics" id="activity-semantics" data-count="18" data-title="Semantics 18 (activity-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#activity-semantics">Semantics 18 (activity-semantics)</a></div>
-  <p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-384-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-384">W,\rho \models activity(id,st,et,attrs)</script>
+  <p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-394-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-394">W,\rho \models activity(id,st,et,attrs)</script>
   holds if and only if:</p>
 <ol>
-<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-385-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-385">id</script> maps to an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-386-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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-386">act = \rho(id) \in Activities</script>
-</li>
-<li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-387-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">(</mo><mi>s</mi><mi>t</mi><mo stretchy="false">)</mo><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-387">\rho(st) \in Times</script> is the activity's start time, that is:
-<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-388-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>s</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-388">startTime(id) = \rho(st)</script>
+<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-395-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-395">id</script> maps to an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-396-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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-396">act = \rho(id) \in Activities</script>
 </li>
-<li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-389-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">(</mo><mi>e</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-389">\rho(et)</script> is the activity's end time, that is: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-390-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>e</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-390">endTime(id) = \rho(et)</script>
+<li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-397-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">(</mo><mi>s</mi><mi>t</mi><mo stretchy="false">)</mo><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-397">\rho(st) \in Times</script> is the activity's start time, that is:
+<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-398-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>s</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-398">startTime(id) = \rho(st)</script>
 </li>
-<li>There exists <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-391-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-391">start,e,a</script> such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-392-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>c</mi><mi>t</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-392">started(start) = (act,e,a)</script>.</li>
-<li>There exists <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-393-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-393">end,e',a'</script> such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-394-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>c</mi><mi>t</mi><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><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-394">ended(end) = (act,e',a')</script>.</li>
-<li>The attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-395-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><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></math></span></span></span><script type="math/tex" id="MathJax-Element-395">match(W,act,attrs)</script>.
+<li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-399-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">(</mo><mi>e</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-399">\rho(et)</script> is the activity's end time, that is: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-400-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>e</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-400">endTime(id) = \rho(et)</script>
+</li>
+<li>There exists <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-401-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-401">start,e,a</script> such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-402-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>c</mi><mi>t</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-402">started(start) = (act,e,a)</script>.</li>
+<li>There exists <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-403-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-403">end,e',a'</script> such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-404-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>c</mi><mi>t</mi><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><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-404">ended(end) = (act,e',a')</script>.</li>
+<li>The attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-405-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><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></math></span></span></span><script type="math/tex" id="MathJax-Element-405">match(W,act,attrs)</script>.
 </li>
 </ol>
 </div>
 </section>
 <section id="agent">
 
-<h4><span class="secno">8.3.3 </span> Agent </h4>
-
-<p>An agent formula is of the form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-396-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-396">agent(id,attrs)</script> where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-397-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-397">id</script> denotes the agent and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-398-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-398">attrs</script> describes additional attributes.
+<h4><span class="secno">4.3.3 </span> Agent </h4>
+
+<p>An agent formula is of the form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-406-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-406">agent(id,attrs)</script> where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-407-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-407">id</script> denotes the agent and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-408-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-408">attrs</script> describes additional attributes.
 </p>
 <div class="semantics" id="agent-semantics" data-count="19" data-title="Semantics 19 (agent-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#agent-semantics">Semantics 19 (agent-semantics)</a></div>
-  <p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-399-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-399">W,\rho \models agent(id,attrs)</script> holds if and only if:
+  <p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-409-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-409">W,\rho \models agent(id,attrs)</script> holds if and only if:
   </p>
   <ol>
-    <li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-400-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-400">id</script> denotes an agent <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-401-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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-401">ag = \rho(id) \in Agents</script>
+    <li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-410-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-410">id</script> denotes an agent <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-411-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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-411">ag = \rho(id) \in Agents</script>
     </li>
-    <li>The attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-402-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</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></math></span></span></span><script type="math/tex" id="MathJax-Element-402">match(W,ag,attrs)</script>.
+    <li>The attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-412-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</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></math></span></span></span><script type="math/tex" id="MathJax-Element-412">match(W,ag,attrs)</script>.
     </li>
   </ol>
 </div>
@@ -2053,215 +2063,215 @@
 </section>
 
 <section id="semantics-of-relations">
-<h3><span class="secno">8.4 </span> Semantics of Relations </h3>
+<h3><span class="secno">4.4 </span> Semantics of Relations </h3>
 
 
 <section id="generation">
-<h4><span class="secno">8.4.1 </span> Generation </h4>
+<h4><span class="secno">4.4.1 </span> Generation </h4>
 
 <p>The generation formula is of the form
-<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-403-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-403">wasGeneratedBy(id,e,a,t,attrs)</script> where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-404-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-404">id</script> is an event identifier,
-<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-405-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-405">e</script> is an entity identifier, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-406-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-406">a</script> is an activity identifier, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-407-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-407">attrs</script> is
-a set of attribute-value pairs, and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-408-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-408">t</script> is a time.
+<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-413-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-413">wasGeneratedBy(id,e,a,t,attrs)</script> where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-414-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-414">id</script> is an event identifier,
+<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-415-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-415">e</script> is an entity identifier, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-416-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-416">a</script> is an activity identifier, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-417-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-417">attrs</script> is
+a set of attribute-value pairs, and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-418-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-418">t</script> is a time.
 </p>
 <div class="semantics" id="generation-semantics" data-count="20" data-title="Semantics 20 (generation-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#generation-semantics">Semantics 20 (generation-semantics)</a></div>
- <p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-409-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-409">W,\rho \models
+ <p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-419-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-419">W,\rho \models
   wasGeneratedBy(id,e,a,t,attrs)</script>  holds if and only if:
 </p><ol>
-<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-410-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-410">id</script> denotes a generation event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-411-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</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></math></span></span></span><script type="math/tex" id="MathJax-Element-411">evt = \rho(id) \in Generations</script>.
-</li>
-<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-412-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-412">e</script> denotes an entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-413-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-413">ent = \rho(e) \in Entities</script>.
+<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-420-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-420">id</script> denotes a generation event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-421-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</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></math></span></span></span><script type="math/tex" id="MathJax-Element-421">evt = \rho(id) \in Generations</script>.
 </li>
-<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-414-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-414">a</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-415-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><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-415">act = \rho(a) \in Activities</script>.
+<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-422-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-422">e</script> denotes an entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-423-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-423">ent = \rho(e) \in Entities</script>.
 </li>
-<li>The event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-416-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-416">evt</script> occurred at time <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-417-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><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-417">\rho(t) \in Times</script>, i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-418-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-418">time(evt) = \rho(t)</script>.
+<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-424-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-424">a</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-425-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><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-425">act = \rho(a) \in Activities</script>.
 </li>
-<li>The activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-419-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-419">act</script> generated <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-420-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-420">ent</script> via <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-421-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-421">evt</script>, i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-422-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>e</mi><mi>v</mi><mi>t</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-422">generated(evt) = (ent,act)</script>.
+<li>The event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-426-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-426">evt</script> occurred at time <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-427-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><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-427">\rho(t) \in Times</script>, i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-428-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-428">time(evt) = \rho(t)</script>.
 </li>
-<li>The attribute values match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-423-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</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></math></span></span></span><script type="math/tex" id="MathJax-Element-423">match(W,evt,attrs)</script>.
+<li>The activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-429-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-429">act</script> generated <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-430-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-430">ent</script> via <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-431-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-431">evt</script>, i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-432-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>e</mi><mi>v</mi><mi>t</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-432">generated(evt) = (ent,act)</script>.
+</li>
+<li>The attribute values match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-433-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</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></math></span></span></span><script type="math/tex" id="MathJax-Element-433">match(W,evt,attrs)</script>.
 </li>
 </ol>
 </div>
 
 </section>
 <section id="use">
-<h4><span class="secno">8.4.2 </span> Use </h4>
-
-<p>The use formula is of the form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-424-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>a</mi><mo stretchy="false">,</mo><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-424">used(id,a,e,t,attrs)</script> where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-425-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-425">id</script>
-denotes an event, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-426-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-426">a</script> is an activity identifier, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-427-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-427">e</script> is an object
-identifier, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-428-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-428">attrs</script> is a set of attribute-value pairs, and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-429-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-429">t</script> is a time.
+<h4><span class="secno">4.4.2 </span> Use </h4>
+
+<p>The use formula is of the form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-434-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>a</mi><mo stretchy="false">,</mo><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-434">used(id,a,e,t,attrs)</script> where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-435-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-435">id</script>
+denotes an event, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-436-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-436">a</script> is an activity identifier, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-437-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-437">e</script> is an object
+identifier, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-438-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-438">attrs</script> is a set of attribute-value pairs, and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-439-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-439">t</script> is a time.
 </p>
 <div class="semantics" id="usage-semantics" data-count="21" data-title="Semantics 21 (usage-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#usage-semantics">Semantics 21 (usage-semantics)</a></div>
-  <p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-430-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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>a</mi><mo stretchy="false">,</mo><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-430">W,\rho \models used(id,a,e,t,attrs)</script> holds if and only if:</p>
+  <p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-440-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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>a</mi><mo stretchy="false">,</mo><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-440">W,\rho \models used(id,a,e,t,attrs)</script> holds if and only if:</p>
   <ol>
 
-<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-431-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-431">id</script> denotes a usage event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-432-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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-432">evt = \rho(id) \in Usages</script>.
-</li>
-<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-433-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-433">a</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-434-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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-434">act = \rho(id) \in Activities</script>.
+<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-441-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-441">id</script> denotes a usage event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-442-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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-442">evt = \rho(id) \in Usages</script>.
 </li>
-<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-435-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-435">e</script> denotes an entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-436-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-436">ent = \rho(e) \in Entities</script>.
+<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-443-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-443">a</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-444-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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-444">act = \rho(id) \in Activities</script>.
 </li>
-<li>The event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-437-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-437">evt</script> occurred at time <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-438-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><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-438">\rho(t) \in Times</script>,
-i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-439-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-439">time(evt) = \rho(t)</script>.
+<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-445-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-445">e</script> denotes an entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-446-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-446">ent = \rho(e) \in Entities</script>.
 </li>
-<li>The activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-440-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-440">act</script> used <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-441-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-441">obj</script> via <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-442-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-442">evt</script>, i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-443-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>e</mi><mi>v</mi><mi>t</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-443">used(evt) = (act,ent)</script>.
+<li>The event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-447-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-447">evt</script> occurred at time <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-448-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><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-448">\rho(t) \in Times</script>,
+i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-449-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-449">time(evt) = \rho(t)</script>.
 </li>
-<li>The attribute values match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-444-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</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></math></span></span></span><script type="math/tex" id="MathJax-Element-444">match(W,evt,attrs)</script>.
+<li>The activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-450-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-450">act</script> used <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-451-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-451">obj</script> via <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-452-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-452">evt</script>, i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-453-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>e</mi><mi>v</mi><mi>t</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-453">used(evt) = (act,ent)</script>.
+</li>
+<li>The attribute values match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-454-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</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></math></span></span></span><script type="math/tex" id="MathJax-Element-454">match(W,evt,attrs)</script>.
 </li></ol></div>
 
 </section>
 <section id="invalidation">
-<h4><span class="secno">8.4.3 </span> Invalidation </h4>
-
-<p>The invalidation formula is of the form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-445-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-445">wasInvalidatedBy(id,e,a,t,attrs)</script> where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-446-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-446">id</script> is an event identifier, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-447-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-447">e</script> is an entity identifier, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-448-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-448">a</script> is an activity identifier, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-449-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-449">attrs</script> is a set of attribute-value pairs, and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-450-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-450">t</script> is a time.</p>
-
-<div class="semantics" id="invalidation-semantics" data-count="22" data-title="Semantics 22 (invalidation-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#invalidation-semantics">Semantics 22 (invalidation-semantics)</a></div> <p>An invalidation formula <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-451-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-451">W,\rho \models
+<h4><span class="secno">4.4.3 </span> Invalidation </h4>
+
+<p>The invalidation formula is of the form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-455-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-455">wasInvalidatedBy(id,e,a,t,attrs)</script> where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-456-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-456">id</script> is an event identifier, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-457-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-457">e</script> is an entity identifier, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-458-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-458">a</script> is an activity identifier, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-459-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-459">attrs</script> is a set of attribute-value pairs, and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-460-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-460">t</script> is a time.</p>
+
+<div class="semantics" id="invalidation-semantics" data-count="22" data-title="Semantics 22 (invalidation-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#invalidation-semantics">Semantics 22 (invalidation-semantics)</a></div> <p>An invalidation formula <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-461-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-461">W,\rho \models
   wasInvalidatedBy(id,e,a,t,attrs)</script> holds  if and only if:</p>
 <ol>
-<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-452-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-452">id</script> denotes an invalidation event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-453-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</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-453">evt = \rho(id) \in Invalidations</script>.
-</li>
-<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-454-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-454">e</script> denotes an entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-455-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-455">ent = \rho(e) \in Entities</script>.
-</li>
-<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-456-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-456">a</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-457-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><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-457">act = \rho(a) \in Activities</script>.
+<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-462-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-462">id</script> denotes an invalidation event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-463-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</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-463">evt = \rho(id) \in Invalidations</script>.
 </li>
-<li>The event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-458-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-458">evt</script> occurred at time <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-459-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><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-459">\rho(t) \in Times</script>,
-  i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-460-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-460">time(evt) = \rho(t)</script>.
+<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-464-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-464">e</script> denotes an entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-465-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-465">ent = \rho(e) \in Entities</script>.
 </li>
-<li>The activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-461-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-461">act</script> invalidated <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-462-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-462">ent</script> via <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-463-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-463">evt</script>, i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-464-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>e</mi><mi>v</mi><mi>t</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-464">invalidated(evt) = (ent,act)</script>.
+<li>[WF] The identifier <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-466-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-466">a</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-467-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><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-467">act = \rho(a) \in Activities</script>.
 </li>
-<li>The attribute values match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-465-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</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></math></span></span></span><script type="math/tex" id="MathJax-Element-465">match(W,evt,attrs)</script>.
+<li>The event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-468-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-468">evt</script> occurred at time <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-469-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><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-469">\rho(t) \in Times</script>,
+  i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-470-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-470">time(evt) = \rho(t)</script>.
+</li>
+<li>The activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-471-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-471">act</script> invalidated <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-472-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-472">ent</script> via <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-473-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-473">evt</script>, i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-474-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>e</mi><mi>v</mi><mi>t</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-474">invalidated(evt) = (ent,act)</script>.
+</li>
+<li>The attribute values match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-475-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</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></math></span></span></span><script type="math/tex" id="MathJax-Element-475">match(W,evt,attrs)</script>.
 </li></ol></div>
 </section>
 
 <section id="association">
 
-<h4><span class="secno">8.4.4 </span> Association </h4>
-
-<p>An association formula has the form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-466-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><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><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-466">wasAssociatedWith(id,a,ag,pl,attrs)</script>.</p>
+<h4><span class="secno">4.4.4 </span> Association </h4>
+
+<p>An association formula has the form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-476-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><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><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-476">wasAssociatedWith(id,a,ag,pl,attrs)</script>.</p>
 
 <div class="semantics" id="association-plan-semantics" data-count="23" data-title="Semantics 23 (association-plan-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#association-plan-semantics">Semantics 23 (association-plan-semantics)</a></div><p>
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-467-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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><mo stretchy="false">,</mo><mi>a</mi><mi>g</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></math></span></span></span><script type="math/tex" id="MathJax-Element-467">W,\rho \models wasAssociatedWith(id,a,ag,pl,attrs)</script> holds if and only if:</p>
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-477-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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><mo stretchy="false">,</mo><mi>a</mi><mi>g</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></math></span></span></span><script type="math/tex" id="MathJax-Element-477">W,\rho \models wasAssociatedWith(id,a,ag,pl,attrs)</script> holds if and only if:</p>
 <ol>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-468-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-468">assoc</script> denotes an association <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-469-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-469">assoc = \rho(id) \in Associations</script>.
-</li>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-470-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-470">a</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-471-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><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-471">act = \rho(a) \in Activities</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-478-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-478">assoc</script> denotes an association <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-479-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-479">assoc = \rho(id) \in Associations</script>.
 </li>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-472-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-472">ag</script> denotes an agent <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-473-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo><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-473">agent = \rho(ag) \in Agents</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-480-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-480">a</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-481-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><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-481">act = \rho(a) \in Activities</script>.
 </li>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-474-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-474">pl</script> denotes a plan <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-475-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>a</mi><mi>n</mi><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>p</mi><mi>l</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>P</mi><mi>l</mi><mi>a</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-475">plan=\rho(pl) \in Plans</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-482-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-482">ag</script> denotes an agent <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-483-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo><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-483">agent = \rho(ag) \in Agents</script>.
 </li>
-<li>The association associates the agent with the activity and plan, i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-476-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>g</mi><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><mi>p</mi><mi>l</mi><mi>a</mi><mi>n</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-476">associatedWith(assoc) = (agent,act,plan)</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-484-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-484">pl</script> denotes a plan <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-485-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>a</mi><mi>n</mi><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>p</mi><mi>l</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>P</mi><mi>l</mi><mi>a</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-485">plan=\rho(pl) \in Plans</script>.
 </li>
-<li>The attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-477-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><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-477">match(W,assoc,attrs)</script>.
+<li>The association associates the agent with the activity and plan, i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-486-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>g</mi><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><mi>p</mi><mi>l</mi><mi>a</mi><mi>n</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-486">associatedWith(assoc) = (agent,act,plan)</script>.
+</li>
+<li>The attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-487-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><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-487">match(W,assoc,attrs)</script>.
 </li></ol></div>
 
 <div class="semantics" id="assocation-semantics" data-count="24" data-title="Semantics 24 (assocation-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#assocation-semantics">Semantics 24 (assocation-semantics)</a></div><p>
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-478-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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><mo stretchy="false">,</mo><mi>a</mi><mi>g</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></math></span></span></span><script type="math/tex" id="MathJax-Element-478">W,\rho \models wasAssociatedWith(id,a,ag,-,attrs)</script> holds if and only if:</p>
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-488-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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><mo stretchy="false">,</mo><mi>a</mi><mi>g</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></math></span></span></span><script type="math/tex" id="MathJax-Element-488">W,\rho \models wasAssociatedWith(id,a,ag,-,attrs)</script> holds if and only if:</p>
 <ol>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-479-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-479">assoc</script> denotes an association <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-480-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-480">assoc = \rho(id) \in Associations</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-489-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-489">assoc</script> denotes an association <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-490-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-490">assoc = \rho(id) \in Associations</script>.
 </li>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-481-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-481">a</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-482-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><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-482">act = \rho(a) \in Activities</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-491-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-491">a</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-492-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><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-492">act = \rho(a) \in Activities</script>.
 </li>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-483-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-483">ag</script> denotes an agent <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-484-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo><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-484">agent = \rho(ag) \in Agents</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-493-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-493">ag</script> denotes an agent <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-494-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo><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-494">agent = \rho(ag) \in Agents</script>.
 </li>
 <li>The association associates the agent with the activity and no
-  plan, i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-485-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>g</mi><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><mi mathvariant="normal">⊥</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-485">associatedWith(assoc) = (agent,act,\bot)</script>.
+  plan, i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-495-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>g</mi><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><mi mathvariant="normal">⊥</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-495">associatedWith(assoc) = (agent,act,\bot)</script>.
 </li>
-<li>The attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-486-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><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-486">match(W,assoc,attrs)</script>.
+<li>The attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-496-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><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-496">match(W,assoc,attrs)</script>.
 </li></ol></div>
 </section>
 <section id="start">
-<h4><span class="secno">8.4.5 </span> Start </h4>
-
-<p>A start formula <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-487-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-487">wasStartedBy(id,a_2,e,a_1,t,attrs)</script> is interpreted as follows:</p>
+<h4><span class="secno">4.4.5 </span> Start </h4>
+
+<p>A start formula <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-497-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-497">wasStartedBy(id,a_2,e,a_1,t,attrs)</script> is interpreted as follows:</p>
 
 <div class="semantics" id="start-semantics" data-count="25" data-title="Semantics 25 (start-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#start-semantics">Semantics 25 (start-semantics)</a></div>
-  <p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-488-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-488">W,\rho \models wasStartedBy(id,a_2,e,a_1,t,attrs)</script> holds if and only if:</p>
+  <p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-498-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-498">W,\rho \models wasStartedBy(id,a_2,e,a_1,t,attrs)</script> holds if and only if:</p>
 <ol>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-489-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-489">id</script> denotes a start event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-490-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>S</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-490">evt = \rho(id) \in Starts</script>.
-</li>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-491-Frame" style="font-size: 100%; "><span class="MathJax_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>a</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-491">a_2</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-492-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>2</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">)</mo><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-492">act_2 = \rho(a_2) \in Activities</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-499-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-499">id</script> denotes a start event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-500-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>S</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-500">evt = \rho(id) \in Starts</script>.
 </li>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-493-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-493">e</script> denotes an entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-494-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-494">ent = \rho(e) \in Entities</script>.
-</li>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-495-Frame" style="font-size: 100%; "><span class="MathJax_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>a</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-495">a_1</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-496-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>1</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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-496">act_1 = \rho(a_1) \in Activities</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-501-Frame" style="font-size: 100%; "><span class="MathJax_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>a</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-501">a_2</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-502-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>2</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">)</mo><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-502">act_2 = \rho(a_2) \in Activities</script>.
 </li>
-<li>The event happened at the start of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-497-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-497">act_2</script>, that is, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-498-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo 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></math></span></span></span><script type="math/tex" id="MathJax-Element-498">\rho(t) = startTime(act_2) = time(evt)</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-503-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-503">e</script> denotes an entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-504-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-504">ent = \rho(e) \in Entities</script>.
 </li>
-<li> The activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-499-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-499">act_1</script> started <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-500-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-500">act_2</script> via entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-501-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-501">ent</script>: that is,
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-502-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-502">started(evt) = (act_2,ent,act_1)</script>.</li>
-<li>The attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-503-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</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></math></span></span></span><script type="math/tex" id="MathJax-Element-503">match(W,evt,attrs)</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-505-Frame" style="font-size: 100%; "><span class="MathJax_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>a</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-505">a_1</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-506-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>1</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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-506">act_1 = \rho(a_1) \in Activities</script>.
+</li>
+<li>The event happened at the start of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-507-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-507">act_2</script>, that is, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-508-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo 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></math></span></span></span><script type="math/tex" id="MathJax-Element-508">\rho(t) = startTime(act_2) = time(evt)</script>.
+</li>
+<li> The activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-509-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-509">act_1</script> started <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-510-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-510">act_2</script> via entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-511-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-511">ent</script>: that is,
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-512-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-512">started(evt) = (act_2,ent,act_1)</script>.</li>
+<li>The attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-513-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</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></math></span></span></span><script type="math/tex" id="MathJax-Element-513">match(W,evt,attrs)</script>.
 </li></ol></div>
 </section>
 <section id="end">
-<h4><span class="secno">8.4.6 </span> End </h4>
-
-<p>An activity end formula <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-504-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-504">wasEndedBy(id,a_2,e,a_1,t,attrs)</script> is interpreted as follows:</p>
+<h4><span class="secno">4.4.6 </span> End </h4>
+
+<p>An activity end formula <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-514-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-514">wasEndedBy(id,a_2,e,a_1,t,attrs)</script> is interpreted as follows:</p>
 
 <div class="semantics" id="end-semantics" data-count="26" data-title="Semantics 26 (end-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#end-semantics">Semantics 26 (end-semantics)</a></div>
 <p>
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-505-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-505">W,\rho \models wasEndedBy(id,a_2,e,a_1,t,attrs)</script> holds if and only
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-515-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-515">W,\rho \models wasEndedBy(id,a_2,e,a_1,t,attrs)</script> holds if and only
   if:</p>
   
 <ol>
-  <li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-506-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-506">id</script> denotes an end event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-507-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>E</mi><mi>n</mi><mi>d</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-507">evt = \rho(id) \in Ends</script>.</li>
-<li> [WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-508-Frame" style="font-size: 100%; "><span class="MathJax_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>a</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-508">a_2</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-509-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>2</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">)</mo><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-509">act_2 = \rho(a_2)\in Activities</script>.</li>
-<li> [WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-510-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-510">e</script> denotes an entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-511-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-511">ent = \rho(e)\in Entities</script>.</li>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-512-Frame" style="font-size: 100%; "><span class="MathJax_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>a</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-512">a_1</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-513-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>1</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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-513">act_1 = \rho(a_1)\in Activities</script>.</li>
-<li>The event happened at the end of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-514-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-514">act_2</script>, that is, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-515-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>e</mi><mi>n</mi><mi>d</mi><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo 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></math></span></span></span><script type="math/tex" id="MathJax-Element-515">\rho(t) = endTime(act_2) = time(evt)</script>.</li>
-<li> The activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-516-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-516">act_1</script> ended <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-517-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-517">act_2</script> via entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-518-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-518">ent</script>: that is,
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-519-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>v</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-519">ended(evt) = (act_2,ent,act_1)</script>.</li>
-<li>The attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-520-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</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></math></span></span></span><script type="math/tex" id="MathJax-Element-520">match(W,evt,attrs)</script>.</li>
+  <li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-516-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-516">id</script> denotes an end event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-517-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>E</mi><mi>n</mi><mi>d</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-517">evt = \rho(id) \in Ends</script>.</li>
+<li> [WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-518-Frame" style="font-size: 100%; "><span class="MathJax_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>a</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-518">a_2</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-519-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>2</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">)</mo><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-519">act_2 = \rho(a_2)\in Activities</script>.</li>
+<li> [WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-520-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-520">e</script> denotes an entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-521-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-521">ent = \rho(e)\in Entities</script>.</li>
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-522-Frame" style="font-size: 100%; "><span class="MathJax_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>a</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-522">a_1</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-523-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>1</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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-523">act_1 = \rho(a_1)\in Activities</script>.</li>
+<li>The event happened at the end of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-524-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-524">act_2</script>, that is, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-525-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>e</mi><mi>n</mi><mi>d</mi><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo 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></math></span></span></span><script type="math/tex" id="MathJax-Element-525">\rho(t) = endTime(act_2) = time(evt)</script>.</li>
+<li> The activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-526-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-526">act_1</script> ended <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-527-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-527">act_2</script> via entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-528-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-528">ent</script>: that is,
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-529-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>v</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-529">ended(evt) = (act_2,ent,act_1)</script>.</li>
+<li>The attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-530-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</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></math></span></span></span><script type="math/tex" id="MathJax-Element-530">match(W,evt,attrs)</script>.</li>
 </ol>
 </div>
 </section>
 
 <section id="attribution">
-<h4><span class="secno">8.4.7 </span> Attribution </h4>
+<h4><span class="secno">4.4.7 </span> Attribution </h4>
 
 <p>
-An attribution formula <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-521-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-521">wasAttributedTo(id,e,ag,attrs)</script> is interpreted as follows:
+An attribution formula <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-531-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-531">wasAttributedTo(id,e,ag,attrs)</script> is interpreted as follows:
 </p>
 <div class="semantics" id="attribution-semantics" data-count="27" data-title="Semantics 27 (attribution-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#attribution-semantics">Semantics 27 (attribution-semantics)</a></div>
   <p>
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-522-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-522">W,\rho \models wasAttributedTo(id,e,ag,attrs)</script>
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-532-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-532">W,\rho \models wasAttributedTo(id,e,ag,attrs)</script>
   holds if and only if:
   </p>
 <ol>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-523-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-523">id</script> denotes an association <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-524-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-524">assoc = \rho(id) \in Associations</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-533-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-533">id</script> denotes an association <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-534-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-534">assoc = \rho(id) \in Associations</script>.
 </li>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-525-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-525">e</script> denotes an entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-526-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-526">ent = \rho(e) \in Entities</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-535-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-535">e</script> denotes an entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-536-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-536">ent = \rho(e) \in Entities</script>.
 </li>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-527-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-527">ag</script> denotes an agent <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-528-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo><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-528">agent = \rho(ag) \in Agents</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-537-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-537">ag</script> denotes an agent <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-538-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo><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-538">agent = \rho(ag) \in Agents</script>.
 </li>
-<li>The entity was attributed to the agent, i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-529-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>s</mi><mi>s</mi><mi>o</mi><mi>c</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>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-529">attributedTo(assoc)
+<li>The entity was attributed to the agent, i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-539-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>s</mi><mi>s</mi><mi>o</mi><mi>c</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>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-539">attributedTo(assoc)
 = (ent,agent)</script>.
 </li>
-<li>The attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-530-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><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-530">match(W,assoc,attrs)</script>.
+<li>The attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-540-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><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-540">match(W,assoc,attrs)</script>.
 </li></ol>
 </div>
 </section>
 
 
 <section id="communication">
-<h4><span class="secno">8.4.8 </span>Communication</h4>
-<p>A communication formula <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-531-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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>2</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></math></span></span></span><script type="math/tex" id="MathJax-Element-531">wasInformedBy(id,a_2,a_2,attrs)</script> is
+<h4><span class="secno">4.4.8 </span>Communication</h4>
+<p>A communication formula <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-541-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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>2</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></math></span></span></span><script type="math/tex" id="MathJax-Element-541">wasInformedBy(id,a_2,a_2,attrs)</script> is
 interpreted as follows: </p>
 <div class="semantics" id="communication-semantics" data-count="28" data-title="Semantics 28 (communication-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#communication-semantics">Semantics 28 (communication-semantics)</a></div>
   <p>
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-532-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-532">W,\rho \models wasInformedBy(id,a_2,a_1,attrs)</script>
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-542-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-542">W,\rho \models wasInformedBy(id,a_2,a_1,attrs)</script>
   holds if and only if:
   </p>
 <ol>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-533-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-533">id</script> denotes a communication <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-534-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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-534">comm = \rho(id) \in Communications</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-543-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-543">id</script> denotes a communication <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-544-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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-544">comm = \rho(id) \in Communications</script>.
 </li>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-535-Frame" style="font-size: 100%; "><span class="MathJax_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>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-535">a_1,a_2</script> denote  activities <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-536-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>1</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">)</mo><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-536">act_1 = \rho(a_1) \in
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-545-Frame" style="font-size: 100%; "><span class="MathJax_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>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-545">a_1,a_2</script> denote  activities <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-546-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>1</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">)</mo><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-546">act_1 = \rho(a_1) \in
 Activities, act_2 = \rho(a_2)\in Activities</script>.
 </li>
-<li>There exist <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-537-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>u</mi><mi>s</mi><mi>e</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-537">gen,use,ent</script> such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-538-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-538">communicated(comm) =
-(act_2,act_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-539-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>n</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-539">generated(gen) = (ent,act_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-540-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>e</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>2</mn></msub><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-540">used(use) = (act_2,ent)</script>.
+<li>There exist <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-547-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>u</mi><mi>s</mi><mi>e</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-547">gen,use,ent</script> such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-548-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-548">communicated(comm) =
+(act_2,act_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-549-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>n</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-549">generated(gen) = (ent,act_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-550-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>e</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>2</mn></msub><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-550">used(use) = (act_2,ent)</script>.
 </li>
-<li>The attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-541-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><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-541">match(W,comm,attrs)</script>.
+<li>The attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-551-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><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-551">match(W,comm,attrs)</script>.
 </li></ol>
 </div>
 </section>
@@ -2269,103 +2279,103 @@
 
 <section id="delegation">
 
-<h4><span class="secno">8.4.9 </span> Delegation </h4>
-
-<p>The <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-542-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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>d</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></math></span></span></span><script type="math/tex" id="MathJax-Element-542">actedOnBehalfOf(id,ag_2,ag_1,act,attrs)</script> relation is interpreted using the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-543-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>s</mi><mi>F</mi><mi>o</mi><mi>r</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-543">ActsFor</script> relation as follows:</p>
+<h4><span class="secno">4.4.9 </span> Delegation </h4>
+
+<p>The <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-552-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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>d</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></math></span></span></span><script type="math/tex" id="MathJax-Element-552">actedOnBehalfOf(id,ag_2,ag_1,act,attrs)</script> relation is interpreted using the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-553-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>s</mi><mi>F</mi><mi>o</mi><mi>r</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-553">ActsFor</script> relation as follows:</p>
 
 <div class="semantics" id="delegation-semantics" data-count="29" data-title="Semantics 29 (delegation-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#delegation-semantics">Semantics 29 (delegation-semantics)</a></div>
   <p>
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-544-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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>d</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></math></span></span></span><script type="math/tex" id="MathJax-Element-544">W,\rho \models actedOnBehalfOf(id,ag_2,ag_1,act,attrs)</script> holds if and only if:</p>
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-554-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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>d</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></math></span></span></span><script type="math/tex" id="MathJax-Element-554">W,\rho \models actedOnBehalfOf(id,ag_2,ag_1,act,attrs)</script> holds if and only if:</p>
 <ol>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-545-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-545">id</script> denotes a delegation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-546-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-546">deleg=\rho(id) \in Delegations</script>.
-</li>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-547-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-547">a</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-548-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><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-548">act=\rho(a) \in Activities</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-555-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-555">id</script> denotes a delegation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-556-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-556">deleg=\rho(id) \in Delegations</script>.
 </li>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-549-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-549">ag_1,ag_2</script> denote agents <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-550-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">)</mo><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-550">agent_1=\rho(ag_1), agent_2=\rho(ag_2) \in Agents</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-557-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-557">a</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-558-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><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-558">act=\rho(a) \in Activities</script>.
 </li>
-<li>The agent <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-551-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-551">agent_2</script> acted for the agent <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-552-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-552">agent_1</script> with respect to
-  the activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-553-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-553">act</script>, i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-554-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>g</mi><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><msub><mi>t</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-554">actedFor(deleg) = (agent_2,agent_1,act)</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-559-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-559">ag_1,ag_2</script> denote agents <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-560-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">)</mo><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-560">agent_1=\rho(ag_1), agent_2=\rho(ag_2) \in Agents</script>.
 </li>
-<li>The attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-555-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>d</mi><mi>e</mi><mi>l</mi><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-555">match(W,deleg,attrs)</script>.
+<li>The agent <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-561-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-561">agent_2</script> acted for the agent <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-562-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>g</mi><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-562">agent_1</script> with respect to
+  the activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-563-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-563">act</script>, i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-564-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>g</mi><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><msub><mi>t</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-564">actedFor(deleg) = (agent_2,agent_1,act)</script>.
+</li>
+<li>The attributes match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-565-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>d</mi><mi>e</mi><mi>l</mi><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-565">match(W,deleg,attrs)</script>.
 </li></ol></div>
 </section>
 
 
 <section id="derivation">
 
-<h4><span class="secno">8.4.10 </span> Derivation </h4>
+<h4><span class="secno">4.4.10 </span> Derivation </h4>
 
 <p>
 Derivation formulas can be of one of two forms:</p>
-<ul><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-556-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>a</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></math></span></span></span><script type="math/tex" id="MathJax-Element-556">wasDerivedFrom(id,e_2,e_1,a,g,u,attrs)</script>, which specifies an
+<ul><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-566-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>a</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></math></span></span></span><script type="math/tex" id="MathJax-Element-566">wasDerivedFrom(id,e_2,e_1,a,g,u,attrs)</script>, which specifies an
   activity, generation and usage event.  For convenience we call this
   a <dfn id="dfn-precise-derivation">precise derivation</dfn>.</li>
-  <li> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-557-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-557">wasDerivedFrom(id,e_2,e_1,-,-,-,attrs)</script>, which does not
+  <li> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-567-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-567">wasDerivedFrom(id,e_2,e_1,-,-,-,attrs)</script>, which does not
   specify an activity, generation and usage event.  For convenience we
   call this an <dfn id="dfn-imprecise-derivation">imprecise derivation</dfn>.</li>
   </ul>
 
 <section id="precise">
-<h5><span class="secno">8.4.10.1 </span> Precise </h5>
-
-<p>A precise derivation formula has the form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-558-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>a</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></math></span></span></span><script type="math/tex" id="MathJax-Element-558">wasDerivedFrom(id,e_2,e_1,a,g,u,attrs)</script>.
+<h5><span class="secno">4.4.10.1 </span> Precise </h5>
+
+<p>A precise derivation formula has the form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-568-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>a</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></math></span></span></span><script type="math/tex" id="MathJax-Element-568">wasDerivedFrom(id,e_2,e_1,a,g,u,attrs)</script>.
 </p>
 <div class="semantics" id="derivation-precise-semantics" data-count="30" data-title="Semantics 30 (derivation-precise-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#derivation-precise-semantics">Semantics 30 (derivation-precise-semantics)</a></div>
-  <p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-559-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-559">W,\rho \models wasDerivedFrom(id,e_2,e_1,act,g,u,attrs)</script> holds if and only if:</p>
+  <p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-569-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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></math></span></span></span><script type="math/tex" id="MathJax-Element-569">W,\rho \models wasDerivedFrom(id,e_2,e_1,act,g,u,attrs)</script> holds if and only if:</p>
 <ol>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-560-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-560">id</script> denotes a derivation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-561-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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-561">deriv = \rho(id) \in Derivations</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-570-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-570">id</script> denotes a derivation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-571-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><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-571">deriv = \rho(id) \in Derivations</script>.
 </li>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-562-Frame" style="font-size: 100%; "><span class="MathJax_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></math></span></span></span><script type="math/tex" id="MathJax-Element-562">e_1,e_2</script> denote entities <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-563-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>1</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><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></math></span></span></span><script type="math/tex" id="MathJax-Element-563">ent_1 = \rho(e_1), ent_2=\rho(e_2)  \in Entities</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-572-Frame" style="font-size: 100%; "><span class="MathJax_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></math></span></span></span><script type="math/tex" id="MathJax-Element-572">e_1,e_2</script> denote entities <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-573-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mn>1</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><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></math></span></span></span><script type="math/tex" id="MathJax-Element-573">ent_1 = \rho(e_1), ent_2=\rho(e_2)  \in Entities</script>.
 </li>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-564-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-564">a</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-565-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><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-565">act = \rho(a) \in Activities</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-574-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-574">a</script> denotes an activity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-575-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><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-575">act = \rho(a) \in Activities</script>.
 </li>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-566-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-566">g</script> denotes a generation event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-567-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>g</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></math></span></span></span><script type="math/tex" id="MathJax-Element-567">gen = \rho(g) \in Generations</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-576-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-576">g</script> denotes a generation event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-577-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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 mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>g</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></math></span></span></span><script type="math/tex" id="MathJax-Element-577">gen = \rho(g) \in Generations</script>.
 </li>
-<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-568-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-568">u</script> denotes a use event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-569-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><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-569">\rho(u) \in Usages</script>.
+<li>[WF] <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-578-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-578">u</script> denotes a use event <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-579-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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">ρ</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><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-579">\rho(u) \in Usages</script>.
 </li>
 <li>The derivation denotes a one-step derivation path linking the
-entities via the activity, generation and use: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-570-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">⋅</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">⋅</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">⋅</mo><mi>u</mi><mi>s</mi><mi>e</mi><mo stretchy="false">⋅</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-570">derivationPath(deriv) =
+entities via the activity, generation and use: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-580-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">⋅</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">⋅</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">⋅</mo><mi>u</mi><mi>s</mi><mi>e</mi><mo stretchy="false">⋅</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-580">derivationPath(deriv) =
 ent_2 \cdot gen \cdot act \cdot use \cdot ent_1</script>.
 </li>
-<li>The attribute values match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-571-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><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-571">match(W,deriv,attrs)</script>.</li>
+<li>The attribute values match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-581-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>t</mi><mi>c</mi><mi>h</mi><mo stretchy="false">(</mo><mi>W</mi><mo stretchy="false">,</mo><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><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-581">match(W,deriv,attrs)</script>.</li>
 </ol>
 </div>
 </section>
 <section id="imprecise">
 
-<h5><span class="secno">8.4.10.2 </span> Imprecise </h5>
+<h5><span class="secno">4.4.10.2 </span> Imprecise </h5>
 <p>
-An imprecise derivation formula has the form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-572-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-572">wasDerivedFrom(id,e_2,e_1,-,-,-,attrs)</script>.</p>
+An imprecise derivation formula has the form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-582-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-582">wasDerivedFrom(id,e_2,e_1,-,-,-,attrs)</script>.</p>
 
 <div class="semantics" id="derivation-imprecise-semantics" data-count="31" data-title="Semantics 31 (derivation-imprecise-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#derivation-imprecise-semantics">Semantics 31 (derivation-imprecise-semantics)</a></div>
-  <p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-573-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><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-573">W,\rho \models wasDerivedFrom(id,e_2,e_1,-,-,-,attrs)</script> holds if and only if:</p>
+  <p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-583-Frame" style="font-size: 100%; "><span class="MathJax_MathConta