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* update
authorjcheney@inf.ed.ac.uk
Fri, 05 Apr 2013 16:04:24 +0100
changeset 6054 30775f148bf9
parent 6053 44629fc9a665
child 6055 12d1a8bad30b
* update
semantics/prov-sem.html
semantics/releases/NOTE-prov-sem-20130430/Overview.html 
--- a/semantics/prov-sem.html	Thu Apr 04 17:08:38 2013 +0100
+++ b/semantics/prov-sem.html	Fri Apr 05 16:04:24 2013 +0100
@@ -987,8 +987,9 @@
 data model (called the <dfn>naive semantics</dfn>), viewing
 PROV-DM statements as atomic formulas in the sense of first-order
 logic, and viewing the constraints and inferences specified in
-PROV-CONSTRAINTS as a first-order theory. It is shown that the
-first-order theory is sound with respect to the naive semantics.
+PROV-CONSTRAINTS as a first-order theory. It is shown that valid PROv
+instances (in the sense of PROV-CONSTRAINTS) correspond to satisfiable
+theories.
 This information may be useful to researchers or users of PROV to
 understand the intended meaning and use of PROV for modeling
 information about the actual history, derivation or evolution of Web
@@ -1068,7 +1069,7 @@
 <h3>Purpose of this document</h3>
 
 </section>
-<p>The PROV-DM and PROV-CONSTRAINTS give motivating examples that
+<p>The PROV-DM and PROV-CONSTRAINTS specifications give motivating examples that
 provide an intuition about the meaning of the constructs.  For some
 concepts, such as use, start, end, generation/invalidation, and
 derivation, the meaning is either obvious or situation-dependent.
@@ -1091,13 +1092,10 @@
 intent behind certain features of PROV, to researchers investigating
 richer forms of reasoning over provenance, or to future efforts
 building upon PROV.  It is intended as an exploration of <b>one</b> semantics for PROV, not a definitive specification of the  <b>only</b>
-semantics of PROV.  We intend to provide an intuitive semantics that satisfies all
-of the constraints on valid PROV instances, which ensures that no
-invalid PROV instance has a model.  The current naive semantics, however, is
-not complete in the sense that some valid PROV instances lack models.</p>
-<div class="note">
-  <p>TODO: Revise this to reflect future improvements in the semantics.</p>
-  </div>
+semantics of PROV.  We provide a semantics that satisfies all
+of the constraints on valid PROV instances, and such that valid PROV
+instances correspond to satisfiable theories: every valid instance has
+a model, and vice versa.</p>
 
 <p> Although it is a work in progress, the naive semantics has some appealing
 properties.  Specifically, it provides a declarative counterpart to
@@ -1107,7 +1105,7 @@
 to keep the specification closer to implementations, although other
 implementations are possible and allowed.  In addition to providing a
 naive semantics, this document shows that the operational
-presentation of PROV validity checking is sound with respect to the
+presentation of PROV validity checking is equivalent to the
 declarative presentation adopted here.  This could help justify
 alternative approaches to validity checking.</p>
 
@@ -1146,10 +1144,7 @@
   including soundness and a weak form of completeness: a PROV
   instance is valid if and only if it has a naive model.</li>
 </ul>
-<div class="note">
-<p>TODO: We would like to say something stronger here, such as a
-  completeness result for naive models, but this will take more work.</p>
-  </div>
+
 
 </section>
 <section id="audience">
@@ -1236,6 +1231,16 @@
 </p>
 </section>
 
+<section>
+<h3> Attributes and Values </h3>
+
+<p>We assume a set $Attributes$ of attribute labels and a set $Values$
+of possible values of attributes.  To allow for the fact that some
+attributes can have undefined or multiple values, we sometimes use the set
+$P(Value)$, that is, the set of sets of values.
+</p> </section>
+
+
 
 <section>
 <h3> Times </h3>
@@ -1244,7 +1249,7 @@
 $Times \subseteq Val$ and $\leq$ is a linear order.
 </p>
 
-
+<!--
 <div class="remark">
 <p>Restricting attention to linearly-ordered times, and imposing this
   order on events, is a simplifying assumption; it is more restrictive than required to model the
@@ -1260,7 +1265,7 @@
   and maximum time, written $min([t_1,t_2]) = t_1$ and $max([t_1,t_2])
   = t_2$ respectively.
 </p>
-
+-->
   
 </section>
   
@@ -1268,15 +1273,6 @@
 
 
   
-<section>
-<h3> Attributes and Values </h3>
-
-<p>We assume a set $Attributes$ of attribute labels and a set $Values$
-of possible values of attributes.  To allow for the fact that some
-attributes can have undefined or multiple values, we sometimes use the set
-$P(Value)$, that is, the set of sets of values.
-</p> </section>
-
 <section id="formulas">
 <h2>Atomic Formulas</h2>
 
@@ -1350,7 +1346,7 @@
   bundles.  Future editions of PROV may incorporate $mentionOf$ or
   other cross-instance assertions, and if so this semantics should be
   generalized in order to provide a rationale for such an
-  extension and to establish the soundness of constaints associated
+  extension and to establish the soundness of constraints associated
   with $mentionOf$. </p>
 </div>
 </section>
@@ -1400,6 +1396,8 @@
 collection of sets, functions and relations.  The components of a
 structure $W$ are given in the rest of the section in
 <em>components</em>, highlighted in boxes.
+
+<div class="note"><p>TODO: Introduce Events</p></div>
 <section>
 
 <h3> Things </h3> 
@@ -1410,7 +1408,8 @@
 </p>
 <div class="component" id="things"><ol>
   <li> a set $Things$ of things</li>
-  <li> a function $lifetime : Things \to Intervals$ from things to intervals</li>
+  <li> a function $lifetime : Things \to P(Times)$ from things to
+  sets of time instants.</li>
   <li>a function $value : Things \times Attributes \times Times \to P(Values)$
 </li>
 </ol>
@@ -1425,7 +1424,7 @@
 and attribute values.  A thing could be a record of fixed
 attribute values; it could be a bear; it could be the Royal Society;
 it could be a transcendental number like $\pi$.  All that matters from
-our point of view is that we know how to map the $Thing$ to its time interval and attribute mapping.
+our point of view is that we know how to map the $Thing$ to its lifetime and attribute mapping.
 </p>
 
 
@@ -1467,7 +1466,7 @@
 
 <div class="component" id="objects">
   <ol><li> a set $Objects$ 
-</li><li> a function $lifetime : Objects \to Intervals$ from objects to time intervals
+</li><li> a function $lifetime : Objects \to P(Times)$ from objects to time intervals
 </li><li> a function $value : Objects \times Attributes \to P(Values)$
 </li></ol>
 </div>
@@ -1499,11 +1498,12 @@
   thing. We assume:</p>
 
 <div class="component" id="entities">
-  <ol><li> a set $Entities \subseteq Objects$ of entities, disjoint from $Activities$ and $Events$ below.
+  <ol><li> a set $Entities \subseteq Objects$ of entities, disjoint from $Activities$ below.
 </li><li> a function $thingOf : Entities \to Things$ that associates
-  each $Entity$ $e$ with a $Thing$, such that for each $t \in
-  lifetime(e)$, and for each attribute $a$ we have $value(e,a)
-  \subseteq value(thingOf(e),a,t)$ and $lifetime(e) \subseteq lifetime(thingOf(e))$.
+  each $Entity$ $e$ with a $Thing$, such that $lifetime(e) \subseteq
+  lifetime(thingOf(e))$ and for each $t \in
+  lifetime(e)$ and for each attribute $a$ we have $value(e,a)
+  \subseteq value(thingOf(e),a,t)$
 </li>
 <!--<li>a relation $SpecializationOf \subseteq Entities \times Entities$
   that is irreflexive and transitive.  Furthermore, if $(e_1,e_2) \in
@@ -1628,12 +1628,12 @@
 
 <p>We consider a set $Influences \subseteq Objects$ which has disjoint
   subsets
-  <em>Events</em> connecting entities and activities,
-  <em>Associations</em> between agents and activities,
-    <em>Attributions</em> between entities and agents,
-  <em>Communications</em> between pairs of activities,
-  <em>Delegations</em> between pairs of agents, and
-  <em>Derivations</em> that describe chains of generation and usage
+  $Events$ connecting entities and activities,
+  $Associations$ between agents and activities,
+    $Attributions$ between entities and agents,
+  $Communications$ between pairs of activities,
+  $Delegations$ between pairs of agents, and
+  $Derivations$ that describe chains of generation and usage
   steps.  These kinds of influences are discussed further below.  Influences are disjoint from entities, activities and agents.
 </p>
 <div class="component" id="influences">
@@ -1654,7 +1654,7 @@
 <section>
 <h5> Events </h5>
 
-<p>An <em>Event</em> is an influence whose lifetime is a single time
+<p>An $Event$ is an influence whose lifetime is a single time
 instant, and relates an activity to an entity (which could be an
 agent).  Events have types including usage, generation, invalidation, starting and ending.  Events are instantaneous.  We introduce:
 </p>
@@ -1686,7 +1686,8 @@
 
 <h5> Associations </h5>
 
-<p>An <em>Association</em> is an influence relating an agent to an activity.  To model associations, we introduce:
+<p>An $Association$ is an influence relating an agent to an activity
+and optional plan.  To model associations, we introduce:
 </p>
 <div class="component" id="associations">
   <p>A set $Associations \subseteq Influences$ with associated
@@ -1698,7 +1699,7 @@
 
 <h5> Attributions </h5>
 
-<p>An <em>Attribution</em> is an influence relating an entity to an agent.  To model associations, we introduce:
+<p>An $Attribution$ is an influence relating an entity to an agent.  To model attributions, we introduce:
 </p>
 <div class="component" id="attributions">
   <p>A set $Attributions \subseteq Influences$ with associated
@@ -1709,12 +1710,12 @@
 </section>
   <section>
   <h5>Communications</h5>
-  <p>A <em>Communication</em> is an influence indicating exchange of
+  <p>A $Communication$ is an influence indicating exchange of
   information between activities.  To model communications, we introduce:
 </p>
 <div class="component" id="communications">
   <p>A set $Communications \subseteq Influences$ with associated
-  function $communicatedBy : Communications \to Activities \times Activities$.
+  function $communicated : Communications \to Activities \times Activities$.
 </p>
   </div>
   
@@ -1722,7 +1723,7 @@
 </section>
   <section>
   <h5>Delegations</h5>
-<p>A <em>Delegation</em> is an influence relating  two agents.  To
+<p>A $Delegation$ is an influence relating  two agents.  To
   model delegations, we introduce:
 </p>
 <div class="component" id="delegations">
@@ -1736,7 +1737,7 @@
   
   <h5> Derivations </h5>
 
-<p>A <em>Derivation</em> is an influence chaining one or more
+<p>A $Derivation$ is an influence chaining one or more
   generation and use steps.  To model derivations, we introduce an
   auxiliary notion of <em>derivation path</em>.  These paths are of the form </p>
 
@@ -1809,7 +1810,7 @@
   <ol>
     <li id="axiom1">
     If $generated(g) = (e,a_1)$ and $used(u) = (a_2,e)$ then there
-    exists $c \in Communications$ such that $communicatedBy(c) = (a_2,a_1)$.
+    exists $c \in Communications$ such that $communicated(c) = (a_2,a_1)$.
     </li>
     <li id="axiom2">
     If $started(start) = (a_2,e,a_1)$ then there exists $gen$ such
@@ -1830,8 +1831,8 @@
     </li>
     <li id="axiom6">
     If $actedFor(deleg) = (ag_2,ag_1,act)$ then there exist
-    $assoc_1,assoc_2,pl_1,pl_2$ such that $associatedWith(assoc_1) = (act,ag_1,pl_1)$
-    and $associatedWith(assoc_2) = (act,ag_2,pl_2)$.
+    $assoc_1,assoc_2,pl_1,pl_2$ such that $associatedWith(assoc_1) = (ag_1,act,pl_1)$
+    and $associatedWith(assoc_2) = (ag_2,act,pl_2)$.
     </li>
     <li id="axiom7">
     If $generated(id) = (e,a)$ then $influenced(id) = (e,a)$.
@@ -1840,7 +1841,7 @@
         If $used(id) = (e,a)$ then $influenced(id) = (e,a)$.
     </li>
     <li id="axiom9">
-            If $communicatedBy(id) = (a_2,a_1)$ then $influenced(id) = (a_2,a_1)$.
+            If $communicated(id) = (a_2,a_1)$ then $influenced(id) = (a_2,a_1)$.
     </li>
     <li id="axiom10">
      If $started(id) = (a_2,e,a_1)$ then $influenced(id) = (a_2,e)$.
@@ -1865,7 +1866,7 @@
     If $actedFor(id) = (ag_2,ag_1)$ then $influenced(id) = (ag_2,ag_1)$.
     </li>
     <li id="axiom17">
-    If $generate(gen) = (e,a) = generated(gen')$ then $gen = gen'$.
+    If $generated(gen) = (e,a) = generated(gen')$ then $gen = gen'$.
     </li>
     <li id="axiom18">
     If $invalidated(inv) = (e,a) = invalidated(inv')$ then $inv=inv'$.
@@ -1877,16 +1878,19 @@
     If $ended(end) = (a,e_1,a')$ and $ended(end') = (a,e_2,a')$ then $end=end'$.
     </li>
     <li id="axiom21">
-    If $started(st) = (a,e)$ then $st \preceq evt$ for all $evt \in events(a)$.
+    If $started(st) = (a,e)$ then $st \preceq evt$ for all $evt \in
+    events(a) - Invalidations$.
     </li>
     <li id="axiom22">
-        If $ended(end) = (a,e,a') $ then $evt \preceq end$ for all $evt \in events(a)$.
+        If $ended(end) = (a,e,a') $ then $evt \preceq end$ for all
+    $evt \in events(a) - Invalidations$.
     </li>
     <li id="axiom23">
     If $generated(gen) = (e,a)$ then $gen \preceq evt$ for all $evt \in events(e)$.
     </li>
     <li id="axiom24">
-        If $invalidated(inv) = (e,a)$ then $evt\preceq inv$ for all $evt \in events(e)$.
+        If $invalidated(inv) = (e,a)$ then $evt\preceq inv$ for all
+    $evt \in events(e)$.
     </li>
     <li id="axiom25">
     For any derivation $deriv$, with path $derivationPath(deriv) = w$,
@@ -1899,30 +1903,91 @@
     \cdot w \cdot e_1$, if $generated(gen_1) = (e_1,a_1)$ and
     $generated(gen_2) = (e_2,a_2)$ then $gen_1 \prec gen_2$.  
     </li>
+    <li id="axiom27">
+    If  $associatedWith(assoc) = (a,ag,pl)$  and $started(start) = (a,e_1,a_1)$ and $invalidated(inv) =
+    (ag,a_2)$ then $start \preceq inv$.
+    </li>
+    <li id="axiom28">
+    If  $associatedWith(assoc) = (a,ag,pl)$  and $generated(gen) =
+    (ag,a_1)$ and $ended(end) = (a,e_2,a_2)$ then $gen \preceq end$.
+    </li>
+    <li id="axiom29">
+    If  $associatedWith(assoc) = (a,ag,pl)$  and $started(start) = (a,e_1,a_1)$ and $ended(end) =
+    (ag,e_2,a_2)$ then $start \preceq end$.
+    </li>
+    <li id="axiom30">
+    If  $associatedWith(assoc) = (a,ag,pl)$  and $started(start) =
+    (ag,e_1,a_1)$ and $ended(end) = (a,e_2,a_2)$ then $start \preceq end$.
+    </li>
+       <li id="axiom31">
+    If $attributedTo(attrib) = (e,ag)$  and $generated(gen_1) =
+    (ag_1,a_1)$ and $generated(gen_2) = (e,a_2)$ then $gen_1 \preceq gen_2$.
+    </li>
+       <li id="axiom32">
+    If $attributedTo(attrib) = (e,ag)$  and $started(start) =
+    (ag_1,e_1,a_1)$ and $generated(gen) = (e,a_2)$ then $start \preceq gen$.
+    </li>
+       <li id="axiom33">
+    If $actedFor(deleg) = (ag_2,ag_1,a)$  and $generated(gen) =
+    (ag_1,a_1)$ and $invalidated(inv) = (ag_2,a_2)$ then $gen \preceq inv$.
+    </li>
+       <li id="axiom34">
+    If $actedFor(deleg) = (ag_2,ag_1,a)$ and $started(start) =
+    (ag_1,e_1,a_1)$ and $ended(end) = (ag_2,e_2,a_2)$ then $start \preceq
+    end$.
+    </li>
+    
     </ol>
 </div>
 
     <p>These properties are called <em>axioms</em>, and they are
   needed to ensure that the PROV-CONSTRAINTS inferences and
   constraints hold in all structures.</p>
+
+<div class="remark">
+  <p> Axioms 21 and 22 do not require that invalidation events
+  originating from an activity follow the activity's start
+  event(s) or precede its end event(s).
+  This is because
+  there is no such constraint in PROV-CONSTRAINTS.  Arguably, there
+  should be a constraint analogous to Constraint 34 that specifies
+  that any invalidation event in which an activity participates must
+  follow the activity's start event(s) and precede its end event(s).
+  </div>
   </section>
 
 <section>
 <h3> Putting it all together </h3>
 
-<p>A <em>structure</em> $W$ is a collection of sets, functions, and relations containing all of the above
-described components.  If we need to talk about the objects or relations of
+<p>A <em>PROV structure</em> $W$ is a collection of sets, functions, and relations containing all of the above
+described components and satisfying all of the associated properties
+and axions.  If we need to talk about the objects or relations of
 more than one structure then we may write $W_1.Objects$, $W_1.Things$,
 etc.; otherwise, to
 decrease notational clutter, when we consider a fixed structure then the names of the sets, relations and functions above refer to the components of that model.
 </p>
 
 
-
+<p>
+Some features of PROV structures are relatively obvious or routine,
+corresponding directly to features of PROV and associated inferences.
+For example, the functions $used, generated, invalidated, started,
+ended$ mapping events to their associated entities or activities, and
+$communicated, associatedWith, attributedTo, actedFor$ associating
+other types of influences with appropriate data.  On the other hand,
+some features are more distinctive, and represent areas where formal
+modeling has been used to guide the development of PROV.  Derivation
+paths are one such distinctive feature; they correspond to an
+intuition that derivations may describe one or multiple generation-use
+steps leading from one entity to another.  Another distinctive feature
+is the use of $Things$, which correspond to changing, real-world
+things, as opposed to $Entities$, which correspond to limited views or
+perspectives on $Things$, with some fixed aspects.  The semantic
+structures of $Things$ and $Entities$ provides a foundation for the
+$alternateOf$ and $specializationOf$ relations.
+</p>
   
-<div class="note">
-  TODO: Highlight the distinctive vs obvious/routine features.
-  </div>
+
 
 </section>
 <section>
@@ -2304,7 +2369,7 @@
 <li>[WF] $a_1,a_2$ denote  activities $act_1 = \rho(a_1) \in
 Activities, act_2 = \rho(a_2)\in Activities$.
 </li>
-<li>There exist $gen,use,ent$ such that $communicatedBy(comm) =
+<li>There exist $gen,use,ent$ such that $communicated(comm) =
 (act_2,act_1)$ and $generated(gen) = (ent,act_1)$ and $used(use) = (act_2,ent)$.
 </li>
 <li>The attributes match: $match(W,comm,attrs)$.
@@ -2429,7 +2494,7 @@
   $W,\rho \models specializationOf(e_1,e_2)$ holds if and only if:</p>
 <ol>
 <li>[WF] Both $e_1$ and $e_2$ are entity identifiers, denoting
-  distinct entities $ent_1
+  entities $ent_1
   = \rho(e_1) \in Entities$ and $ent_2 = \rho(e_2) \in Entities$.
 </li>
 <!--<li>$(ent_1,ent_2) \in SpecializationOf$.-->
@@ -2439,8 +2504,8 @@
 </li>
 <li>For each attribute $attr$ we have $value(ent_1,attr) \supseteq value(ent_2,attr)$.</li>
 <li>At least one of these inclusions is strict: that is, either
-  $lifetime(ent_1) \subsetneq lifetime(ent_2)$ or for some $attr$ we
- value(ent_1,attr) \supsetneq value(ent_2,attr)$.
+  $lifetime(ent_1) \subsetneq lifetime(ent_2)$ or for some $attr$ we have
+ $value(ent_1,attr) \supsetneq value(ent_2,attr)$.
 </li>
 </ol>
 </div>
@@ -2462,7 +2527,8 @@
 The last criterion allows $ent_1$ to have more defined attributes than
 $ent_2$, but they must include the attributes defined by $ent_2$.  Two
   different entities that have the same attributes can also be related
-  by specialization.  
+  by specialization.  The fifth criterion (indirectly) ensures that
+  specialization is irreflexive.
 </p>
 </div>
 </section>
@@ -2537,7 +2603,7 @@
   </div>
 
   <div class="remark">
-    <p> Although times are linearly ordered, the time ordering is
+    <p> The ordering of time values associated to events is
     unrelated to the event ordering. For example:</p>
     <pre>
 entity(e)
@@ -2725,7 +2791,7 @@
 \end{array}$</div>
 <div class="proof">
   <p>This follows from the semantics of generation, association, and
-  attribution, by <a href="#axiom5">Axiom 5</a></p>
+  attribution, by <a href="#axiom5">Axiom 5</a>.</p>
   </div>
   
 <div class="inference" number="14" id="delegation-inference">$\begin{array}[t]{l}
@@ -2737,7 +2803,7 @@
 \exists id_1,pl_1,id_2,pl_2.~wasAssociatedWith(id_1,a,ag_1,pl_1,[]) \wedge wasAssociatedWith(id_2,a,ag_2,pl_2,[])
 \end{array}$</div>
 <div class="proof">
-  <p>This follows from the semantics of association and delegation, by <a href="#axiom6">Axiom 6</a></p>
+  <p>This follows from the semantics of association and delegation, by <a href="#axiom6">Axiom 6</a>.</p>
   </div>
 <div class="inference" number="15" id="influence-inference"><ol><li>$\begin{array}[t]{l}
 \forall id,e,a,t,attrs.~
@@ -2909,10 +2975,12 @@
   $entity(e_1,attrs)$ holds, we know that $v \in value(ent_1,att)$.
   Thus $v \in value(ent_2,att)$ since $value(ent_2,att) \supseteq value(ent_1,att)$.  Since
   this is the case for all attribute-value pairs in $attrs$, and since
-  $e_2$ obviously denotes an entity, we can conclude $W,\rho \models entity(e,attrs$).
+  $e_2$ obviously denotes an entity, we can conclude $W,\rho \models entity(e_2,attrs)$.
   </p>
 </div>
-
+<div class="todo"> <p> How do we know $e_2$ has
+  generation/invalidation events?  May need axiom.</p>
+  </div>
 <section>
 <h2>Constraints</h2>
 <section>
@@ -3347,7 +3415,10 @@
 
 
 <div class="proof">
-  <p> ???
+  <p> This follows from <a href="#axiom23">Axiom 23</a> and the fact
+  that if $e_2$ specializes $e_1$ then all of the events of the $e_2$
+  are events of $e_1$.  Thus, the generation of $e_1$ precedes all
+  events of $e_2$.
   </p>
   </div>
 
@@ -3361,7 +3432,10 @@
 \end{array}$</div>
 
 <div class="proof">
-  <p> ???
+  <p> This follows from <a href="#axiom23">Axiom 24</a> and the fact
+  that if $e_2$ specializes $e_1$ then all of the events of the $e_2$
+  are events of $e_1$.  Thus, the invalidation of $e_1$ follows all
+  events of $e_2$.
   </p>
   </div>
 
@@ -3396,7 +3470,8 @@
 \end{array}$</li></ol></div>
 
 <div class="proof">
-  <p>TODO - arbitrary axioms.
+  <p>The four parts follow from <a href="#axiom27">Axiom 27</a> through
+  <a href="#axiom30">Axiom 30</a> respectively.
   </p>
   </div>
 
@@ -3417,7 +3492,8 @@
 \end{array}$</li></ol></div>
 
 <div class="proof">
-  <p>TODO - arbitrary axioms.
+  <p>These properties follow from <a href="#axiom31">Axiom 31</a> and
+  <a href="#axiom 32">Axiom 32</a>.
   </p>
   </div>
 
@@ -3438,7 +3514,8 @@
 \end{array}$</li></ol></div>
 
 <div class="proof">
-  <p>TODO - arbitrary axioms.
+  <p>These properties follow from <a href="#axiom33">Axiom 33</a> and
+  <a href="#axiom 34">Axiom 34</a>.
   </p>
   </div>
 
@@ -3629,8 +3706,8 @@
 \end{array}$</div>
 <div class="proof">
   <p>This follows from the fact that in the semantics of
-  $specializationOf$, the two entities denoted by the first and second
-  arguments are required to be distinct.
+  $specializationOf$, the requirement that one of the inclusions is
+  strict implies that the two entities cannot be the same.
   </p>
   </div>
 <div class="constraint" number="53" id="impossible-property-overlap"><p>For each $r$  and  $s \in \{ used, wasGeneratedBy, wasInvalidatedBy, wasStartedBy, wasEndedBy, wasInformedBy, wasAttributedTo, wasAssociatedWith, actedOnBehalfOf\}$ such that $r$  and  $s$ are different relation names, the following constraint holds:</p>$\begin{array}[t]{l}
@@ -3642,7 +3719,7 @@
 False
 \end{array}$</div>
 <div class="proof">
-  <p>This follows from the assumption that the different classes of
+  <p>This follows from the assumption that the different kinds of
   influences are disjoint sets, characterized by their types.
   </p>
   </div>
@@ -3705,8 +3782,9 @@
 <p>Our main soundness result is:</p>
 
 <div class="theorem" id="soundness-theorem">
-  <p> Let $W$ be a PROV model, that is, a structure satisfying all of
-  the axioms</p>.
+  <p> Let $W$ be a PROV structure, that is, a structure providing all
+  of the components above and satisfying all of
+  the axioms.</p>
   <ol>
   <li>If $I$ is an instance and $W \models I$ and $I'$ is obtained from $I$ by applying one
   of the PROV inferences, then $W \models I'$.</li>
@@ -3724,7 +3802,8 @@
   or uniqueness constraint steps: if $I$ is in normal form then we are
   done. If $I$ is not in normal form then if an inference is  applicable, then use part 1; if a uniqueness constraint is
   applicable, then from $W \models I$ the uniqueness constraint cannot
-  fail on $I$ and $W \models I'$.</p>
+  fail on $I$ and $W \models I'$, as argued for the key constraints in
+  the previous section.</p>
 <p>
   For part 3, the arguments are as
   in the previous section for each constraint. </p>
@@ -3745,13 +3824,13 @@
   <p> Let $I$ be a valid PROV instance that is in normal form.
   We define a structure $M(I)$ as follows, by giving the sets,
   functions and relations specified in the components in <a
-  href="#structures">Section 3</a>.</p>
+  href="#structures">Section 3</a>, and finally verifying that the
+  axioms hold.</p>
 
   <section>
 <h4>Sets</h4>
   <p> The sets of structure $M(I)$ are: </p>
-
-  \[
+\[
   \begin{eqnarray*}
   Entities &=& \{id \mid entity(id,attrs) \in I\}\\
   Plans &=& \{pl \mid wasAssociatedWith(id,ag,act,pl,attrs) \in I, pl
@@ -3821,16 +3900,30 @@
 events in which $e$ participated.</p>
 \[
 \begin{eqnarray*}
-events(e) &=& \{id \mid used(id,a,e,t,attrs) \in I\}\\
+events'(e) &=& \{id \mid used(id,a,e,t,attrs) \in I\}\\
 &\cup& \{id \mid wasGeneratedBy(id,e,a,t,attrs) \in I\}\\
 &\cup& \{id \mid wasInvalidatedBy(id,e,a,t,attrs) \in I\}\\
 &\cup& \{id \mid wasStartedBy(id,a,e,a',t,attrs) \in I\}\\
 &\cup& \{id \mid wasEndedBy(id,a,e,a',t,attrs) \in I\}\\
+events(e) &=& events'(e) \cup \bigcup_{specializationOf(e',e) \in I} events'(e')\\
 lifetime(e) &=& \{time(e) \mid evt \in events(e)\}\\
-value(e,a) &=& \{v \mid entity(e,attrs) \in I, a=v \in attrs\}\\
+value'(e,a) &=& \{v \mid entity(e,attrs) \in I, (a=v) \in attrs\}
+\quad (a \neq uniq)\\
+value'(e,uniq) &=&\{ uniq_{e}\}\\
+value(e,a) &=&  value'(e) \cup \bigcup_{specializationOf(e',e) \in I} value'(e')\\
 thingOf(e) &=& [e]_\equiv
 \end{eqnarray*}
 \]
+<p>Above, we introduce a fresh attribute name $uniq$, not already in
+use in $I$, along with a fresh value $e$ and for each entity $e$ we
+add an attribute-value pair $uniq=uniq_e$ to $values(e,uniq)$.  This
+construction ensures that if an entity is a specialization of another
+in $I$ then the specialization relationship will hold in $M(I)$.  We
+also define the set of all events involved in $e$ as the set of events
+immediately involved in $e$ or any specialization of $e$.  Similarly,
+the values of attributes of $e$ are those immediately declared for $e$
+along with those of any specialization.
+</p>
 <p> Similarly, for $Things$, we
 employ an auxiliary function $events:Things \to P(Events)$ that collects the set of all
 events in which one of the entities constituting the thing participated.</p>
@@ -3857,10 +3950,11 @@
   endTime(id) &=& t_2 \text{ where } activity(a,t_1,t_2,attrs) \in
   I\\
   & \text{ or } wasEndedBy(end,a,e,a',t_2,attrs) \in I\\
+  \end{eqnarray*}
 \]
   <p>Note that the above definition is deterministic because the start
   and end times in $activity$ and $wasStartedBy/wasEndedBy$ statements
-  must be equal</p>.
+  must be equal.</p>
 <p> The function $time$ mapping $Events$ to their $Times$ is defined
   as follows:
   </p>
@@ -3888,7 +3982,7 @@
   I\\
  attributedTo(id) &=& (e,ag) \text{ where } wasAttributedTo(id,e,ag,attrs) \in I\\
 actedFor (id) &=& (ag_2,ag_1,act) \text{ where } actedOnBehalfOf(id,ag_2,ag_1,act,attrs) \in I\\
-  communicatedBy(id) &=& (a_2,a_1) \text{ where }
+  communicated(id) &=& (a_2,a_1) \text{ where }
   wasInformedBy(id,a_2,a_1,attrs)\in I\\
   derivationPath(id) &=& e_2\cdot g \cdot a \cdot u \cdot e_1 \text{ where } wasDerivedFrom(id,e_2,e_1,a,g,u,attrs) \in I\\
   derivationPath(id) &=& e_2\cdot g_{id} \cdot a_{id} \cdot u_{id} \cdot e_1 \text{ where } wasDerivedFrom(id,e_2,e_1,-,-,-,attrs) \in I\\
@@ -3899,13 +3993,7 @@
   imprecise derivations, we generate additional activities, generations
   and usages linking $e_2$ to $e_1$.
   </p>
-  <div class="remark">
-
-    <p> We explicitly add activities, generations, and usages to
-  ensure that we can form derivation paths for imprecise derivation
-  formulas.
-    </p>
-    </div>
+
 </section>
 <section>
 <h4>Relations</h4>
@@ -3931,7 +4019,54 @@
     side-conditions in the components are satisfied.  As noted above,
     the disjointness constraints are satisfied by construction. </p>
 
-    </section>
+<p>For each axiom we give the corresponding justification:</p>
+
+<ol><li>
+  Axiom 1 follows because $I$ is normalized with respect to Inference 6.
+  </li>
+<li>
+Axioms 2 and 3 follow because $I$ is normalized with respect to
+  Inference 9 and 10 respectively.
+  </li><li>
+Axioms 5 and 6 follow because $I$ is normalized with respect to
+  Inference 13 and 14 respectively.
+  </li><li>
+Axioms 7 through 16 follow because $I$ is normalized with respect to
+  Inference 15.
+  </li><li>
+Axioms 17 through 20 follow because $I$ is normalized with respect to
+  uniqueness constraints 24 through 27.
+  </li><li>
+  Axiom 21 follows because constraints 30, 31, 33, 34 ensure that a
+  start event for an activity precedes any other start, end, usage or
+  generation events involving that activity.
+  </li>
+  <li>
+  Axiom 22 follows because constraints 30, 32, 33, 34 ensure that an
+  end event for an activity follows any other events involving that activity.
+  </li>
+  <li>
+  Axiom 23 follows because constraints 34, 36, 37, 39 ensure that a
+  generation event for an entity precedes any other events involving that entity.
+  </li>
+  <li>
+  Axiom 24 follows because constraints 36, 38, 40, 43, 44 ensure that an
+  invalidation event for an entity  follows any other generation,
+  usage, or invalidation events involving
+  that entity.
+  </li>
+  <li>Axiom 25 follows from constraint 41.</li>
+  <li> Axiom 26 follows from constraint 42 and from the fact that the
+  event ordering constraint graph $G_I$ associated with a valid
+  instance $I$ cannot have any cycles involving a strict precedence
+  edge.
+  </li>
+  <li> Axioms 27 through 30 follow from Constraint 47.</li>
+  <li> Axioms 31 and 32 follow from Constraint 48.</li>
+  <li> Axioms 33 and 34 follow from Constraint 49.</li>
+    
+  </ol>
+  </section>
     
 <section>
 <h4>Main results</h4>
@@ -3942,9 +4077,9 @@
     satisfies all of the inferences and constraints.  Thus, a form of
     completeness holds: every valid PROV instance has a model.</p>
 
-<div class="theorem">
+<div class="theorem" id="weak-completeness-theorem">
   <p>Suppose $J$ is a valid PROV instance and $I$ is a normal form for
-  $J$.  Then $M(I) \models J$</p>
+  $J$.  Then $M(I) \models J$.</p>
   </div>
   <div class="proof">
     <p>
@@ -3960,7 +4095,7 @@
     semantics and the definition of the construction of $M$.  </p>
 
     <p> To conclude, we need to show that if $J$ is not in normal
-    form, and norma,izes to $I$, then $M(I) \models J$.  WE can prove
+    form, and normalizes to $I$, then $M(I) \models J$.  We can prove
     this by induction on the length of the sequence of normalization
     steps.  The base case, when $J = I$, is established already.
     Suppose $J$ normalizes in $n+1$ steps and we can perform one
@@ -3981,14 +4116,14 @@
     attrs_2)\}$, where $S$ is a unifying substitution making $S(a_i) =
     S(b_i)$ for each $i \in \{1,\ldots,n\}$.  Since $M(I) \models J'$,
     we must have $M(I),\rho \models J'$  for some $\rho$, and therefore
-    we must also have that $M(I),\rho \models S(J_0)$ and $M(I)\rho \models r(id,S(a_1),\ldots,S(a_n),attrs_1\cup
+    we must also have that $M(I),\rho \models S(J_0)$ and $M(I),\rho \models r(id,S(a_1),\ldots,S(a_n),attrs_1\cup
     attrs_2)$.  We can extend $\rho$ to a valuation $\rho'$ such that
-    $M(I),\rho' \models S(x_1) = x_i \wedge \cdots \wedge S(x_k) =
-    x_k$ where $dom(S) = \{x_1,\ldots,x_k\}$.
+    $M(I),\rho' \models S(x_1) = x_1 \wedge \cdots \wedge S(x_k) =
+    x_k$ where $dom(S) = \{x_1,\ldots,x_k\}$.  Also, 
     $M(I),\rho' \models J_0$ and $M(I),\rho' \models r(id,a_1,\ldots,a_n,attrs_1\cup
     attrs_2)$.  Moreover, since $S$ is a unifier, we also have $M(I),\rho' \models r(id,b_1,\ldots,b_n,attrs_1\cup
     attrs_2)$.  Finally, since we can always remove attributes from an
-    atomic formula without danaging its satisfiability, we can
+    atomic formula without damaging its satisfiability, we can
     conclude that $M(I),\rho' \models r(id,a_1,\ldots,a_n,attrs_1)
     \wedge r(id,b_1,\ldots,b_n,    attrs_2)$.  To conclude, we have
     shown $M(I) \models J_0 \cup \{ r(id,a_1,\ldots,a_n,attrs_1)
--- a/semantics/releases/NOTE-prov-sem-20130430/Overview.html	Thu Apr 04 17:08:38 2013 +0100
+++ b/semantics/releases/NOTE-prov-sem-20130430/Overview.html	Fri Apr 05 16:04:24 2013 +0100
@@ -779,7 +779,7 @@
 span.MathJax_MathML {display: inline}
 div.MathJax_MathML {display: block}
 .MathJax_mmlExBox {display: block; overflow: hidden; height: 1px; width: 60ex; padding: 0; border: 0; margin: 0}
-</style><!--[if lt IE 9]><script src='http://www.w3.org/2008/site/js/html5shiv.js'></script><![endif]--></head> 
+</style><!--[if lt IE 9]><script src='http://www.w3.org/2008/site/js/html5shiv.js'></script><![endif]--><!--[if lt IE 9]><script src='http://www.w3.org/2008/site/js/html5shiv.js'></script><![endif]--></head> 
   <body><div id="MathJax_Message" style="display: none; "></div><div class="head">
   <p>
     
@@ -849,8 +849,9 @@
 data model (called the <dfn id="dfn-naive-semantics">naive semantics</dfn>), viewing
 PROV-DM statements as atomic formulas in the sense of first-order
 logic, and viewing the constraints and inferences specified in
-PROV-CONSTRAINTS as a first-order theory. It is shown that the
-first-order theory is sound with respect to the naive semantics.
+PROV-CONSTRAINTS as a first-order theory. It is shown that valid PROv
+instances (in the sense of PROV-CONSTRAINTS) correspond to satisfiable
+theories.
 This information may be useful to researchers or users of PROV to
 understand the intended meaning and use of PROV for modeling
 information about the actual history, derivation or evolution of Web
@@ -939,7 +940,7 @@
       
     
   
-</section><section id="toc"><h2 class="introductory">Table of Contents</h2><ul class="toc"><li class="tocline"><a href="#introduction" class="tocxref"><span class="secno">1. </span>Introduction</a><ul class="toc"><li class="tocline"><a href="#purpose" class="tocxref"><span class="secno">1.1 </span>Purpose of this document</a></li><li class="tocline"><a href="#structure-of-this-document" class="tocxref"><span class="secno">1.2 </span>Structure of this document</a></li><li class="tocline"><a href="#audience" class="tocxref"><span class="secno">1.3 </span> Audience </a></li></ul></li><li class="tocline"><a href="#basics" class="tocxref"><span class="secno">2. </span> Basics </a><ul class="toc"><li class="tocline"><a href="#identifiers" class="tocxref"><span class="secno">2.1 </span> Identifiers </a></li><li class="tocline"><a href="#times" class="tocxref"><span class="secno">2.2 </span> Times </a></li><li class="tocline"><a href="#attributes-and-values" class="tocxref"><span class="secno">2.3 </span> Attributes and Values </a></li><li class="tocline"><a href="#formulas" class="tocxref"><span class="secno">2.4 </span>Atomic Formulas</a></li><li class="tocline"><a href="#first-order-formulas" class="tocxref"><span class="secno">2.5 </span>First-Order Formulas</a></li></ul></li><li class="tocline"><a href="#structures" class="tocxref"><span class="secno">3. </span> Structures and Interpretations </a><ul class="toc"><li class="tocline"><a href="#things-1" class="tocxref"><span class="secno">3.1 </span> Things </a></li><li class="tocline"><a href="#objects-1" class="tocxref"><span class="secno">3.2 </span> Objects </a><ul class="toc"><li class="tocline"><a href="#entities-1" class="tocxref"><span class="secno">3.2.1 </span> Entities </a><ul class="toc"><li class="tocline"><a href="#plans-1" class="tocxref"><span class="secno">3.2.1.1 </span> Plans </a></li><li class="tocline"><a href="#collections-1" class="tocxref"><span class="secno">3.2.1.2 </span>Collections</a></li></ul></li><li class="tocline"><a href="#activities-1" class="tocxref"><span class="secno">3.2.2 </span> Activities </a></li><li class="tocline"><a href="#agents-1" class="tocxref"><span class="secno">3.2.3 </span> Agents </a></li><li class="tocline"><a href="#influences-1" class="tocxref"><span class="secno">3.2.4 </span> Influences </a><ul class="toc"><li class="tocline"><a href="#events-1" class="tocxref"><span class="secno">3.2.4.1 </span> Events </a></li><li class="tocline"><a href="#associations-1" class="tocxref"><span class="secno">3.2.4.2 </span> Associations </a></li><li class="tocline"><a href="#attributions-1" class="tocxref"><span class="secno">3.2.4.3 </span> Attributions </a></li><li class="tocline"><a href="#communications-1" class="tocxref"><span class="secno">3.2.4.4 </span>Communications</a></li><li class="tocline"><a href="#delegations-1" class="tocxref"><span class="secno">3.2.4.5 </span>Delegations</a></li><li class="tocline"><a href="#derivations-1" class="tocxref"><span class="secno">3.2.4.6 </span> Derivations </a></li></ul></li></ul></li><li class="tocline"><a href="#additional-axioms" class="tocxref"><span class="secno">3.3 </span>Additional axioms</a></li><li class="tocline"><a href="#putting-it-all-together" class="tocxref"><span class="secno">3.4 </span> Putting it all together </a></li><li class="tocline"><a href="#interpretations" class="tocxref"><span class="secno">3.5 </span> Interpretations </a></li></ul></li><li class="tocline"><a href="#semantics" class="tocxref"><span class="secno">4. </span> Semantics </a><ul class="toc"><li class="tocline"><a href="#satisfaction" class="tocxref"><span class="secno">4.1 </span> Satisfaction </a></li><li class="tocline"><a href="#attribute-matching" class="tocxref"><span class="secno">4.2 </span> Attribute matching </a></li><li class="tocline"><a href="#semantics-of-element-formulas" class="tocxref"><span class="secno">4.3 </span> Semantics of Element Formulas </a><ul class="toc"><li class="tocline"><a href="#entity" class="tocxref"><span class="secno">4.3.1 </span> Entity </a></li><li class="tocline"><a href="#activity" class="tocxref"><span class="secno">4.3.2 </span> Activity </a></li><li class="tocline"><a href="#agent" class="tocxref"><span class="secno">4.3.3 </span> Agent </a></li></ul></li><li class="tocline"><a href="#semantics-of-relations" class="tocxref"><span class="secno">4.4 </span> Semantics of Relations </a><ul class="toc"><li class="tocline"><a href="#generation" class="tocxref"><span class="secno">4.4.1 </span> Generation </a></li><li class="tocline"><a href="#use" class="tocxref"><span class="secno">4.4.2 </span> Use </a></li><li class="tocline"><a href="#invalidation" class="tocxref"><span class="secno">4.4.3 </span> Invalidation </a></li><li class="tocline"><a href="#association" class="tocxref"><span class="secno">4.4.4 </span> Association </a></li><li class="tocline"><a href="#start" class="tocxref"><span class="secno">4.4.5 </span> Start </a></li><li class="tocline"><a href="#end" class="tocxref"><span class="secno">4.4.6 </span> End </a></li><li class="tocline"><a href="#attribution" class="tocxref"><span class="secno">4.4.7 </span> Attribution </a></li><li class="tocline"><a href="#communication" class="tocxref"><span class="secno">4.4.8 </span>Communication</a></li><li class="tocline"><a href="#delegation" class="tocxref"><span class="secno">4.4.9 </span> Delegation </a></li><li class="tocline"><a href="#derivation" class="tocxref"><span class="secno">4.4.10 </span> Derivation </a><ul class="toc"><li class="tocline"><a href="#precise" class="tocxref"><span class="secno">4.4.10.1 </span> Precise </a></li><li class="tocline"><a href="#imprecise" class="tocxref"><span class="secno">4.4.10.2 </span> Imprecise </a></li></ul></li><li class="tocline"><a href="#influence" class="tocxref"><span class="secno">4.4.11 </span>Influence</a></li><li class="tocline"><a href="#specialization" class="tocxref"><span class="secno">4.4.12 </span> Specialization </a></li><li class="tocline"><a href="#alternate" class="tocxref"><span class="secno">4.4.13 </span> Alternate </a></li><li class="tocline"><a href="#membership" class="tocxref"><span class="secno">4.4.14 </span> Membership </a></li></ul></li><li class="tocline"><a href="#semantics-of-auxiliary-formulas" class="tocxref"><span class="secno">4.5 </span>Semantics of Auxiliary Formulas</a><ul class="toc"><li class="tocline"><a href="#precedes-and-strictly-precedes" class="tocxref"><span class="secno">4.5.1 </span>Precedes and Strictly Precedes</a></li><li class="tocline"><a href="#notnull" class="tocxref"><span class="secno">4.5.2 </span>notNull</a></li><li class="tocline"><a href="#typeof" class="tocxref"><span class="secno">4.5.3 </span>typeOf</a></li></ul></li></ul></li><li class="tocline"><a href="#theory" class="tocxref"><span class="secno">5. </span> Inferences and Constraints </a><ul class="toc"><li class="tocline"><a href="#inferences" class="tocxref"><span class="secno">5.1 </span>Inferences</a></li><li class="tocline"><a href="#constraints" class="tocxref"><span class="secno">5.2 </span>Constraints</a><ul class="toc"><li class="tocline"><a href="#uniqueness-constraints" class="tocxref"><span class="secno">5.2.1 </span>Uniqueness constraints</a></li><li class="tocline"><a href="#ordering-constraints" class="tocxref"><span class="secno">5.2.2 </span>Ordering constraints</a></li><li class="tocline"><a href="#typing-constraints" class="tocxref"><span class="secno">5.2.3 </span>Typing constraints</a></li><li class="tocline"><a href="#impossibility-constraints" class="tocxref"><span class="secno">5.2.4 </span>Impossibility constraints</a></li></ul></li></ul></li><li class="tocline"><a href="#soundness-completeness" class="tocxref"><span class="secno">6. </span>Soundness and Completeness</a><ul class="toc"><li class="tocline"><a href="#soundness" class="tocxref"><span class="secno">6.1 </span>Soundness</a></li><li class="tocline"><a href="#completeness" class="tocxref"><span class="secno">6.2 </span>Weak Completeness</a><ul class="toc"><li class="tocline"><a href="#sets" class="tocxref"><span class="secno">6.2.1 </span>Sets</a></li><li class="tocline"><a href="#functions" class="tocxref"><span class="secno">6.2.2 </span>Functions</a></li><li class="tocline"><a href="#relations" class="tocxref"><span class="secno">6.2.3 </span>Relations</a></li><li class="tocline"><a href="#axioms-1" class="tocxref"><span class="secno">6.2.4 </span>Axioms</a></li><li class="tocline"><a href="#main-results" class="tocxref"><span class="secno">6.2.5 </span>Main results</a></li></ul></li></ul></li><li class="tocline"><a href="#acknowledgements" class="tocxref"><span class="secno">A. </span>Acknowledgements</a></li><li class="tocline"><a href="#references" class="tocxref"><span class="secno">B. </span>References</a><ul class="toc"><li class="tocline"><a href="#informative-references" class="tocxref"><span class="secno">B.1 </span>Informative references</a></li></ul></li></ul></section>
+</section><section id="toc"><h2 class="introductory">Table of Contents</h2><ul class="toc"><li class="tocline"><a href="#introduction" class="tocxref"><span class="secno">1. </span>Introduction</a><ul class="toc"><li class="tocline"><a href="#purpose" class="tocxref"><span class="secno">1.1 </span>Purpose of this document</a></li><li class="tocline"><a href="#structure-of-this-document" class="tocxref"><span class="secno">1.2 </span>Structure of this document</a></li><li class="tocline"><a href="#audience" class="tocxref"><span class="secno">1.3 </span> Audience </a></li></ul></li><li class="tocline"><a href="#basics" class="tocxref"><span class="secno">2. </span> Basics </a><ul class="toc"><li class="tocline"><a href="#identifiers" class="tocxref"><span class="secno">2.1 </span> Identifiers </a></li><li class="tocline"><a href="#attributes-and-values" class="tocxref"><span class="secno">2.2 </span> Attributes and Values </a></li><li class="tocline"><a href="#times" class="tocxref"><span class="secno">2.3 </span> Times </a></li><li class="tocline"><a href="#formulas" class="tocxref"><span class="secno">2.4 </span>Atomic Formulas</a></li><li class="tocline"><a href="#first-order-formulas" class="tocxref"><span class="secno">2.5 </span>First-Order Formulas</a></li></ul></li><li class="tocline"><a href="#structures" class="tocxref"><span class="secno">3. </span> Structures and Interpretations </a><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>
 
 
 
@@ -983,7 +984,7 @@
 <h3><span class="secno">1.1 </span>Purpose of this document</h3>
 
 </section>
-<p>The PROV-DM and PROV-CONSTRAINTS give motivating examples that
+<p>The PROV-DM and PROV-CONSTRAINTS specifications give motivating examples that
 provide an intuition about the meaning of the constructs.  For some
 concepts, such as use, start, end, generation/invalidation, and
 derivation, the meaning is either obvious or situation-dependent.
@@ -1006,13 +1007,10 @@
 intent behind certain features of PROV, to researchers investigating
 richer forms of reasoning over provenance, or to future efforts
 building upon PROV.  It is intended as an exploration of <b>one</b> semantics for PROV, not a definitive specification of the  <b>only</b>
-semantics of PROV.  We intend to provide an intuitive semantics that satisfies all
-of the constraints on valid PROV instances, which ensures that no
-invalid PROV instance has a model.  The current naive semantics, however, is
-not complete in the sense that some valid PROV instances lack models.</p>
-<div class="note"><div class="note-title"><span>Note</span></div><div class="">
-  <p>TODO: Revise this to reflect future improvements in the semantics.</p>
-  </div></div>
+semantics of PROV.  We provide a semantics that satisfies all
+of the constraints on valid PROV instances, and such that valid PROV
+instances correspond to satisfiable theories: every valid instance has
+a model, and vice versa.</p>
 
 <p> Although it is a work in progress, the naive semantics has some appealing
 properties.  Specifically, it provides a declarative counterpart to
@@ -1022,7 +1020,7 @@
 to keep the specification closer to implementations, although other
 implementations are possible and allowed.  In addition to providing a
 naive semantics, this document shows that the operational
-presentation of PROV validity checking is sound with respect to the
+presentation of PROV validity checking is equivalent to the
 declarative presentation adopted here.  This could help justify
 alternative approaches to validity checking.</p>
 
@@ -1061,10 +1059,7 @@
   including soundness and a weak form of completeness: a PROV
   instance is valid if and only if it has a naive model.</li>
 </ul>
-<div class="note"><div class="note-title"><span>Note</span></div><div class="">
-<p>TODO: We would like to say something stronger here, such as a
-  completeness result for naive models, but this will take more work.</p>
-  </div></div>
+
 
 </section>
 <section id="audience">
@@ -1151,15 +1146,25 @@
 </p>
 </section>
 
+<section id="attributes-and-values">
+<h3><span class="secno">2.2 </span> Attributes and Values </h3>
+
+<p>We assume a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-7-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-7">Attributes</script> of attribute labels and a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-8-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-8">Values</script>
+of possible values of attributes.  To allow for the fact that some
+attributes can have undefined or multiple values, we sometimes use the set
+<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-9-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-9">P(Value)</script>, that is, the set of sets of values.
+</p> </section>
+
+
 
 <section id="times">
-<h3><span class="secno">2.2 </span> Times </h3>
-
-<p>We assume an ordered set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-7-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi><mo stretchy="false">,</mo><mo stretchy="false">≤</mo><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-7">(Times,\leq)</script> of time instants, where
-<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-8-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>V</mi><mi>a</mi><mi>l</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-8">Times \subseteq Val</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-9-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">≤</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-9">\leq</script> is a linear order.
+<h3><span class="secno">2.3 </span> Times </h3>
+
+<p>We assume an ordered set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-10-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi><mo stretchy="false">,</mo><mo stretchy="false">≤</mo><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-10">(Times,\leq)</script> of time instants, where
+<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-11-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>V</mi><mi>a</mi><mi>l</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-11">Times \subseteq Val</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-12-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">≤</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-12">\leq</script> is a linear order.
 </p>
 
-
+<!--
 <div class="remark">
 <p>Restricting attention to linearly-ordered times, and imposing this
   order on events, is a simplifying assumption; it is more restrictive than required to model the
@@ -1170,12 +1175,12 @@
   </p>
   </div>
   
-<p>We also consider a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-10-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-10">Intervals</script> of subsets of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-11-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-11">Times</script>  of the
-  form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-12-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><mi>t</mi><mo stretchy="false">∣</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">≤</mo><mi>t</mi><mo stretchy="false">≤</mo><msub><mi>t</mi><mn>2</mn></msub><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-12">[t_1,t_2] = \{t \mid t_1 \leq t \leq t_2\}</script>.  Every interval has a minimum
-  and maximum time, written <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-13-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">=</mo><msub><mi>t</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-13">min([t_1,t_2]) = t_1</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-14-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>a</mi><mi>x</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">=</mo><msub><mi>t</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-14">max([t_1,t_2])
-  = t_2</script> respectively.
+<p>We also consider a set $Intervals$ of subsets of $Times$  of the
+  form $[t_1,t_2] = \{t \mid t_1 \leq t \leq t_2\}$.  Every interval has a minimum
+  and maximum time, written $min([t_1,t_2]) = t_1$ and $max([t_1,t_2])
+  = t_2$ respectively.
 </p>
-
+-->
   
 </section>
   
@@ -1183,26 +1188,17 @@
 
 
   
-<section id="attributes-and-values">
-<h3><span class="secno">2.3 </span> Attributes and Values </h3>
-
-<p>We assume a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-15-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-15">Attributes</script> of attribute labels and a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-16-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-16">Values</script>
-of possible values of attributes.  To allow for the fact that some
-attributes can have undefined or multiple values, we sometimes use the set
-<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-17-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-17">P(Value)</script>, that is, the set of sets of values.
-</p> </section>
-
 <section id="formulas">
 <h3><span class="secno">2.4 </span>Atomic Formulas</h3>
 
-<p>The following atomic formulas correspond to the statements of PROV-DM.  We assume that definitions 1-4 of PROV-CONSTRAINTS have been applied in order to expand all optional parameters; thus, we use uniform notation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-18-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-18">r(id,a_1,\ldots,a_n)</script> instead of the semicolon notation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-19-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">;</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-19">r(id;a_1,\ldots,a_n)</script>.
+<p>The following atomic formulas correspond to the statements of PROV-DM.  We assume that definitions 1-4 of PROV-CONSTRAINTS have been applied in order to expand all optional parameters; thus, we use uniform notation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-13-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-13">r(id,a_1,\ldots,a_n)</script> instead of the semicolon notation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-14-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">;</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-14">r(id;a_1,\ldots,a_n)</script>.
 </p>
 <p>Each parameter is either an identifier, a constant (e.g. a time or
   other literal value in an attribute list), or a null symbol "-".
-  Placeholder symbols <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-20-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">−</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-20">-</script> can only appear in the specified arguments
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-21-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>l</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-21">pl</script> in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-22-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-22">wasAssociatedWith</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-23-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo stretchy="false">,</mo><mi>g</mi><mo stretchy="false">,</mo><mi>u</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-23">a,g,u</script> in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-24-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mi>a</mi><mi>s</mi><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>r</mi><mi>o</mi><mi>m</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-24">wasDerivedFrom</script>, as shown in the grammar below.
+  Placeholder symbols <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-15-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">−</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-15">-</script> can only appear in the specified arguments
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-16-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>l</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-16">pl</script> in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-17-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-17">wasAssociatedWith</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-18-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo stretchy="false">,</mo><mi>g</mi><mo stretchy="false">,</mo><mi>u</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-18">a,g,u</script> in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-19-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mi>a</mi><mi>s</mi><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>r</mi><mi>o</mi><mi>m</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-19">wasDerivedFrom</script>, as shown in the grammar below.
 </p>
-<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-25-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi>a</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>i</mi><mi>c</mi><mi mathvariant="normal">_</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>u</mi><mi>l</mi><mi>a</mi></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">::=</mo></mrow></mtd><mtd><mi>e</mi><mi>l</mi><mi>e</mi><mi>m</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi mathvariant="normal">_</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>u</mi><mi>l</mi><mi>a</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>r</mi><mi>e</mi><mi>l</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi mathvariant="normal">_</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>u</mi><mi>l</mi><mi>a</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>a</mi><mi>u</mi><mi>x</mi><mi>i</mi><mi>l</mi><mi>i</mi><mi>a</mi><mi>r</mi><mi>y</mi><mi mathvariant="normal">_</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>u</mi><mi>l</mi><mi>a</mi></mtd></mtr><mtr><mtd><mi>e</mi><mi>l</mi><mi>e</mi><mi>m</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi mathvariant="normal">_</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>u</mi><mi>l</mi><mi>a</mi></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">::=</mo></mrow></mtd><mtd><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>a</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>s</mi><mi>t</mi><mo stretchy="false">,</mo><mi>e</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>r</mi><mi>e</mi><mi>l</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi mathvariant="normal">_</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>u</mi><mi>l</mi><mi>a</mi></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">::=</mo></mrow></mtd><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>S</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>E</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>T</mi><mi>o</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>a</mi><mi>c</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>O</mi><mi>n</mi><mi>B</mi><mi>e</mi><mi>h</mi><mi>a</mi><mi>l</mi><mi>f</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mi>f</mi><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo 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stretchy="false">|</mo></mrow></mtd><mtd><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>h</mi><mi>a</mi><mi>d</mi><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>a</mi><mi>u</mi><mi>x</mi><mi>i</mi><mi>l</mi><mi>i</mi><mi>a</mi><mi>r</mi><mi>y</mi><mi mathvariant="normal">_</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>u</mi><mi>l</mi><mi>a</mi></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">::=</mo></mrow></mtd><mtd><mi>x</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">strictlyPrecedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>y</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>x</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>y</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">,</mo><mi>t</mi><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi></mtd><mtd><mo stretchy="false">::=</mo></mtd><mtd><mo stretchy="false">[</mo><mi>a</mi><mi>t</mi><mi>t</mi><msub><mi>r</mi><mn>1</mn></msub><mo stretchy="false">=</mo><msub><mi>v</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><msub><mi>r</mi><mi>n</mi></msub><mo stretchy="false">=</mo><msub><mi>v</mi><mi>n</mi></msub><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mi>t</mi><mi>y</mi></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">::=</mo></mrow></mtd><mtd><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>a</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>y</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>E</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-20">
 \newcommand{\precedes}{~\mathrel{\textrm{precedes}}~}
 \newcommand{\strictlyPrecedes}{~\mathrel{\textrm{strictlyPrecedes}}~}
   \begin{array}{rcl}
@@ -1245,28 +1241,28 @@
 </script>
 
 <div class="remark">
-  <p>We include the standard PROV collection types (<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-26-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-26">Collection</script> and
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-27-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-27">EmptyCollection</script>) and the membership relation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-28-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mi>a</mi><mi>d</mi><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-28">hadMember</script>; however,
+  <p>We include the standard PROV collection types (<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-21-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-21">Collection</script> and
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-22-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-22">EmptyCollection</script>) and the membership relation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-23-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mi>a</mi><mi>d</mi><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-23">hadMember</script>; however,
   we do not model dictionaries or the insertion or deletion relations
   in PROV-DICTIONARY [<cite><a class="bibref" href="#bib-PROV-DICTIONARY">PROV-DICTIONARY</a></cite>], since these are not part
   of the PROV recommendations.  If these features are incorporated
   into future standards, their semantics (and the soundness of the
   associated constraints) should be modeled.
-  We omit the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-29-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>r</mi><mi>o</mi><mi>v</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-29">prov</script> prefixes from the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-30-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-30">Collection</script> and
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-31-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-31">EmptyCollection</script> types.
+  We omit the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-24-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>r</mi><mi>o</mi><mi>v</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-24">prov</script> prefixes from the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-25-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-25">Collection</script> and
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-26-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-26">EmptyCollection</script> types.
   </p>
 
   <p>As stated in the Introduction, we do not explicitly model
   bundles or PROV documents; however, each instance can be viewed as
   a set of formulas and can be modeled separately.  The semantics of
   the standard features of PROV can be defined without talking about
-  multiple instances; however, the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-32-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-32">mentionOf</script> relation in
+  multiple instances; however, the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-27-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-27">mentionOf</script> relation in
   PROV-LINKS [<cite><a class="bibref" href="#bib-PROV-LINKS">PROV-LINKS</a></cite>] is intended to support linking across
-  bundles.  Future editions of PROV may incorporate <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-33-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-33">mentionOf</script> or
+  bundles.  Future editions of PROV may incorporate <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-28-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-28">mentionOf</script> or
   other cross-instance assertions, and if so this semantics should be
   generalized in order to provide a rationale for such an
-  extension and to establish the soundness of constaints associated
-  with <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-34-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-34">mentionOf</script>. </p>
+  extension and to establish the soundness of constraints associated
+  with <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-29-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-29">mentionOf</script>. </p>
 </div>
 </section>
 
@@ -1275,7 +1271,7 @@
 
 <p>We also consider the usual connectives and quantifiers of
 first-order logic [<cite><a class="bibref" href="#bib-Logic">Logic</a></cite>].</p>
-<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-35-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="italic">ϕ</mi></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">::=</mo></mrow></mtd><mtd><mi>a</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>i</mi><mi>c</mi><mi mathvariant="normal">_</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>u</mi><mi>l</mi><mi>a</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>T</mi><mi>r</mi><mi>u</mi><mi>e</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>x</mi><mo stretchy="false">=</mo><mi>y</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi mathvariant="normal">¬</mi><mtext>&nbsp;</mtext><mi mathvariant="italic">ϕ</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><msub><mi mathvariant="italic">ϕ</mi><mn>1</mn></msub><mo stretchy="false">∧</mo><msub><mi mathvariant="italic">ϕ</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><msub><mi mathvariant="italic">ϕ</mi><mn>1</mn></msub><mo stretchy="false">∨</mo><msub><mi mathvariant="italic">ϕ</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><msub><mi mathvariant="italic">ϕ</mi><mn>1</mn></msub><mo stretchy="false">⇒</mo><msub><mi mathvariant="italic">ϕ</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi mathvariant="normal">∀</mi><mi>x</mi><mo>.</mo><mi mathvariant="italic">ϕ</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi mathvariant="normal">∃</mi><mi>x</mi><mo>.</mo><mi mathvariant="italic">ϕ</mi></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-35">
+<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-30-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="italic">ϕ</mi></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">::=</mo></mrow></mtd><mtd><mi>a</mi><mi>t</mi><mi>o</mi><mi>m</mi><mi>i</mi><mi>c</mi><mi mathvariant="normal">_</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>u</mi><mi>l</mi><mi>a</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>T</mi><mi>r</mi><mi>u</mi><mi>e</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi>x</mi><mo stretchy="false">=</mo><mi>y</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi mathvariant="normal">¬</mi><mtext>&nbsp;</mtext><mi mathvariant="italic">ϕ</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><msub><mi mathvariant="italic">ϕ</mi><mn>1</mn></msub><mo stretchy="false">∧</mo><msub><mi mathvariant="italic">ϕ</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><msub><mi mathvariant="italic">ϕ</mi><mn>1</mn></msub><mo stretchy="false">∨</mo><msub><mi mathvariant="italic">ϕ</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><msub><mi mathvariant="italic">ϕ</mi><mn>1</mn></msub><mo stretchy="false">⇒</mo><msub><mi mathvariant="italic">ϕ</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi mathvariant="normal">∀</mi><mi>x</mi><mo>.</mo><mi mathvariant="italic">ϕ</mi></mtd></mtr><mtr><mtd></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow></mtd><mtd><mi mathvariant="normal">∃</mi><mi>x</mi><mo>.</mo><mi mathvariant="italic">ϕ</mi></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-30">
 \begin{array}{rcl}
   \phi &{::=}& atomic\_formula\\
 & | & True\\
@@ -1310,45 +1306,48 @@
 <section id="structures">
 <!--OddPage--><h2><span class="secno">3. </span> Structures and Interpretations </h2>
 
-<p> In this section we define mathematical structures <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-36-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-36">W</script> that can be used to
+<p> In this section we define mathematical structures <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-31-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-31">W</script> that can be used to
 interpret PROV formulas and instances.  A structure consists of a
 collection of sets, functions and relations.  The components of a
-structure <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-37-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-37">W</script> are given in the rest of the section in
+structure <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-32-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-32">W</script> are given in the rest of the section in
 <em>components</em>, highlighted in boxes.
-</p><section id="things-1">
+
+</p><div class="note"><div class="note-title"><span>Note</span></div><div class=""><p>TODO: Introduce Events</p></div></div>
+<section id="things-1">
 
 <h3><span class="secno">3.1 </span> Things </h3> 
 
 <p><em>Things</em>  is a set of things in the situation being modeled.  Each thing has a lifetime during which it exists and attributes whose values can change over time.
 </p>
-<p>To model this, a structure <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-38-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-38">W</script> includes:
+<p>To model this, a structure <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-33">W</script> includes:
 </p>
 <div class="component" id="things" data-count="1" data-title="Component 1 (things)"><div class="ruleTitle"><a class="internalDFN" href="#things">Component 1 (things)</a></div><ol>
-  <li> a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-39-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-39">Things</script> of things</li>
-  <li> a function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-40-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">:</mo><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi><mo stretchy="false">→</mo><mi>I</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-40">lifetime : Things \to Intervals</script> from things to intervals</li>
-  <li>a function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-41-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">:</mo><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi><mo stretchy="false">→</mo><mi>P</mi><mo stretchy="false">(</mo><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-41">value : Things \times Attributes \times Times \to P(Values)</script>
+  <li> a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-34-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-34">Things</script> of things</li>
+  <li> a function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-35-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">:</mo><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi><mo stretchy="false">→</mo><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-35">lifetime : Things \to P(Times)</script> from things to
+  sets of time instants.</li>
+  <li>a function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-36-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">:</mo><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi><mo stretchy="false">→</mo><mi>P</mi><mo stretchy="false">(</mo><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-36">value : Things \times Attributes \times Times \to P(Values)</script>
 </li>
 </ol>
 </div>
 <p>
-The range of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-42-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-42">value</script> is the set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-43-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-43">P(Values)</script>, indicating that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-44-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-44">value</script>
-is essentially a multi-valued function that returns a set of values (possibly empty).    When <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-45-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-45">value(x,a,t) =
-\emptyset</script>, we say that attribute <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-46-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-46">a</script> is undefined for <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-47-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-47">x</script> at time <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-48-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-48">t</script>.</p>
+The range of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-37-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-37">value</script> is the set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-38-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-38">P(Values)</script>, indicating that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-39-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-39">value</script>
+is essentially a multi-valued function that returns a set of values (possibly empty).    When <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-40-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-40">value(x,a,t) =
+\emptyset</script>, we say that attribute <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-41-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-41">a</script> is undefined for <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-42-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-42">x</script> at time <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-43-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-43">t</script>.</p>
 
 <p>Note that this description does not say what the structure of a
-<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-49-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-49">Thing</script> is, only how it may be described in terms of its lifetime
+<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-44-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-44">Thing</script> is, only how it may be described in terms of its lifetime
 and attribute values.  A thing could be a record of fixed
 attribute values; it could be a bear; it could be the Royal Society;
-it could be a transcendental number like <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-50-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="italic">π</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-50">\pi</script>.  All that matters from
-our point of view is that we know how to map the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-51-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-51">Thing</script> to its time interval and attribute mapping.
+it could be a transcendental number like <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-45-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="italic">π</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-45">\pi</script>.  All that matters from
+our point of view is that we know how to map the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-46-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-46">Thing</script> to its lifetime and attribute mapping.
 </p>
 
 
 <p>The identity of a Thing is not observable through its attributes or
-lifetime, so it is possible for two different <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-52-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-52">Things</script> to be indistinguishable by their
-attribute values and lifetime.  That is, if the set of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-53-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>T</mi><mn>0</mn></msub><mo stretchy="false">,</mo><msub><mi>T</mi><mn>1</mn></msub><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-53">Things = \{T_0,T_1\}</script> and the attributes are
-specified as <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-54-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><msub><mi>T</mi><mn>0</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><msub><mi>T</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-54">value(T_0,a,t) = value(T_1,a,t)</script> for each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-55-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo stretchy="false">∈</mo><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-55">t\in
-Times</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-56-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo stretchy="false">∈</mo><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-56">a \in Attributes</script>, this does not imply that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-57-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>0</mn></msub><mo stretchy="false">=</mo><msub><mi>T</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-57">T_0 = T_1</script>.
+lifetime, so it is possible for two different <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-47-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-47">Things</script> to be indistinguishable by their
+attribute values and lifetime.  That is, if the set of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-48-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><msub><mi>T</mi><mn>0</mn></msub><mo stretchy="false">,</mo><msub><mi>T</mi><mn>1</mn></msub><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-48">Things = \{T_0,T_1\}</script> and the attributes are
+specified as <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-49-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><msub><mi>T</mi><mn>0</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><msub><mi>T</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-49">value(T_0,a,t) = value(T_1,a,t)</script> for each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-50-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><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-50">t\in
+Times</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-51-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo stretchy="false">∈</mo><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-51">a \in Attributes</script>, this does not imply that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-52-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>0</mn></msub><mo stretchy="false">=</mo><msub><mi>T</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-52">T_0 = T_1</script>.
 </p>
 
 
@@ -1358,17 +1357,17 @@
 <h3><span class="secno">3.2 </span> Objects </h3>
 
 
-<p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-58-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-58">Things</script> are things in the world that have attributes that
-  can change over time.  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-59-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-59">Things</script> may not have distinguishing features
+<p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-53-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-53">Things</script> are things in the world that have attributes that
+  can change over time.  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-54-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-54">Things</script> may not have distinguishing features
   that are readily observable and permanent.  In PROV, we do not talk
-  explicitly about <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-60-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-60">Things</script>, but instead we talk about various objects
+  explicitly about <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-55-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><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-55">Things</script>, but instead we talk about various objects
   that have discrete, fixed features,  and relationships among these
-  objects. Some objects, called <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-61-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-61">Entities</script>, are associated with
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-62-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-62">Things</script>, and their fixed attributes need to match those of the
-  associated <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-63-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-63">Thing</script> during their common lifetime.  Others correspond
+  objects. Some objects, called <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-56-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>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-56">Entities</script>, are associated with
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-57-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-57">Things</script>, and their fixed attributes need to match those of the
+  associated <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-58-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-58">Thing</script> during their common lifetime.  Others correspond
   to agents, activities, or identifiable interactions among them.</p>  
 
-<p>In this section, we detail the different subsets of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-64-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-64">Objects</script>, and
+<p>In this section, we detail the different subsets of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-59">Objects</script>, and
 give disjointness constraints and associated functions.  Generally, these constraints are necessary to validate
 disjointness constraints from PROV-CONSTRAINTS [<cite><a class="bibref" href="#bib-PROV-CONSTRAINTS">PROV-CONSTRAINTS</a></cite>].
 </p>
@@ -1381,29 +1380,29 @@
 </p>
 
 <div class="component" id="objects" data-count="2" data-title="Component 2 (objects)"><div class="ruleTitle"><a class="internalDFN" href="#objects">Component 2 (objects)</a></div>
-  <ol><li> a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-65-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-65">Objects</script> 
-</li><li> a function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-66-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">:</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi><mo stretchy="false">→</mo><mi>I</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-66">lifetime : Objects \to Intervals</script> from objects to time intervals
-</li><li> a function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-67-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">:</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>s</mi><mo stretchy="false">→</mo><mi>P</mi><mo stretchy="false">(</mo><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-67">value : Objects \times Attributes \to P(Values)</script>
+  <ol><li> a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-60-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-60">Objects</script> 
+</li><li> a function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-61-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">:</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi><mo stretchy="false">→</mo><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-61">lifetime : Objects \to P(Times)</script> from objects to time intervals
+</li><li> a function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-62-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">:</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>s</mi><mo stretchy="false">→</mo><mi>P</mi><mo stretchy="false">(</mo><mi>V</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-62">value : Objects \times Attributes \to P(Values)</script>
 </li></ol>
 </div>
 
-<p>Intuitively, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-68-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-68">lifetime(e)</script> is the time interval during which object
-<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-69-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-69">e</script> exists.  The set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-70-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-70">value(e,a)</script> is the set of values of attribute <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-71-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-71">a</script> during the object's lifetime.
+<p>Intuitively, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-63-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-63">lifetime(e)</script> is the time interval during which object
+<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-64-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-64">e</script> exists.  The set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-65-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-65">value(e,a)</script> is the set of values of attribute <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-66-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-66">a</script> during the object's lifetime.
 </p>
 
-<p>As with <em>Things</em>, the range of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-72-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-72">value</script> is sets of values,
-making <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-73-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-73">value</script> effectively a multivalued function.  It is also
+<p>As with <em>Things</em>, the range of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-67-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-67">value</script> is sets of values,
+making <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-68-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-68">value</script> effectively a multivalued function.  It is also
 possible to have two different objects that are indistinguishable by
 their attributes and time intervals.  Objects are not things, and the
-sets of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-74-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-74">Objects</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-75-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-75">Things</script> are disjoint; however, certain objects,
+sets of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-69-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-69">Objects</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-70-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-70">Things</script> are disjoint; however, certain objects,
 namely entities, are associated with things.
 </p>
 
 <div class="remark">
   <p>
-  Disjointness between <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-76-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-76">Objects</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-77-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-77">Things</script> is not necessary but is
+  Disjointness between <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-71-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-71">Objects</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-72-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-72">Things</script> is not necessary but is
   assumed in order to avoid confusion between the different categories
-  (time-varying <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-78-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-78">Things</script> vs fixed <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-79-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-79">Objects</script>).
+  (time-varying <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-73-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-73">Things</script> vs fixed <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-74-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-74">Objects</script>).
   </p>
   </div>
   
@@ -1414,11 +1413,12 @@
   thing. We assume:</p>
 
 <div class="component" id="entities" data-count="3" data-title="Component 3 (entities)"><div class="ruleTitle"><a class="internalDFN" href="#entities">Component 3 (entities)</a></div>
-  <ol><li> a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-80-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-80">Entities \subseteq Objects</script> of entities, disjoint from <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-81-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-81">Activities</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-82-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-82">Events</script> below.
-</li><li> a function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-83-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">:</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">→</mo><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-83">thingOf : Entities \to Things</script> that associates
-  each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-84-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-84">Entity</script> <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-85-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-85">e</script> with a <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-86-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-86">Thing</script>, such that for each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-87-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo stretchy="false">∈</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-87">t \in
-  lifetime(e)</script>, and for each attribute <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-88-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-88">a</script> we have <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-89-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">⊆</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-89">value(e,a)
-  \subseteq value(thingOf(e),a,t)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-90-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">⊆</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-90">lifetime(e) \subseteq lifetime(thingOf(e))</script>.
+  <ol><li> a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-75-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-75">Entities \subseteq Objects</script> of entities, disjoint from <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-76-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-76">Activities</script> below.
+</li><li> a function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-77-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">:</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">→</mo><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-77">thingOf : Entities \to Things</script> that associates
+  each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-78-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-78">Entity</script> <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-79-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-79">e</script> with a <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-80-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-80">Thing</script>, such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-81-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">⊆</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-81">lifetime(e) \subseteq
+  lifetime(thingOf(e))</script> and for each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-82-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo stretchy="false">∈</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-82">t \in
+  lifetime(e)</script> and for each attribute <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-83-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-83">a</script> we have <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-84-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">⊆</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-84">value(e,a)
+  \subseteq value(thingOf(e),a,t)</script>
 </li>
 <!--<li>a relation $SpecializationOf \subseteq Entities \times Entities$
   that is irreflexive and transitive.  Furthermore, if $(e_1,e_2) \in
@@ -1437,44 +1437,44 @@
 
 <div class="remark"><p> Although both entities and things can have
   undefined or multiple attribute values, their meaning is slightly
-  different: for a thing, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-91-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-91">value(x,a,t) = \emptyset</script> means that the
-  attribute <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-92-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-92">a</script> has no value at time <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-93-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-93">t</script>, whereas for an entity,
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-94-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-94">value(x,a) = \emptyset</script> only means that the thing associated to
-  entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-95-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-95">x</script> need not have a
-  fixed value for <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-96-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-96">a</script> during the lifetime of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-97-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-97">x</script>.  This does not imply
-  that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-98-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-98">value(thingOf(e),a,t) = \emptyset</script> when <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-99-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo stretchy="false">∈</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-99">t \in lifetime(e)</script>.
+  different: for a thing, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-85-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-85">value(x,a,t) = \emptyset</script> means that the
+  attribute <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-86-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-86">a</script> has no value at time <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-87-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-87">t</script>, whereas for an entity,
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-88-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-88">value(x,a) = \emptyset</script> only means that the thing associated to
+  entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-89-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-89">x</script> need not have a
+  fixed value for <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-90-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-90">a</script> during the lifetime of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-91-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-91">x</script>.  This does not imply
+  that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-92-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-92">value(thingOf(e),a,t) = \emptyset</script> when <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-93-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo stretchy="false">∈</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-93">t \in lifetime(e)</script>.
   </p>
 
   <p>Furthermore, all of the attribute values of the entity must
   be present in the associated thing throughout the lifetime of the
-  entity.  For example, suppose <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-100-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-100">value(thingOf(e),a,t)</script> is <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-101-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-101">\{1\}</script> at
-  some time in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-102-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-102">lifetime(e)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-103-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><mn>2</mn><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-103">value(thingOf(e),a,t') = \{2\}</script> at
-  some other time <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-104-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>t</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-104">t'</script>.  Then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-105-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-105">value(e,a)</script> must be <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-106-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-106">\emptyset</script> because
+  entity.  For example, suppose <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-94-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-94">value(thingOf(e),a,t)</script> is <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-95-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-95">\{1\}</script> at
+  some time in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-96-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-96">lifetime(e)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-97-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><mn>2</mn><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-97">value(thingOf(e),a,t') = \{2\}</script> at
+  some other time <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-98-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>t</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-98">t'</script>.  Then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-99-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-99">value(e,a)</script> must be <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-100-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-100">\emptyset</script> because
   there is no other set of values that is simultaneously contained in
-  both <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-107-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-107">\{1\}</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-108-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>2</mn><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-108">\{2\}</script>.  </p> </div>
+  both <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-101-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-101">\{1\}</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-102-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">{</mo><mn>2</mn><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-102">\{2\}</script>.  </p> </div>
 
 
 
 <div class="remark">
   <p>
-  In the above description of how <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-109-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-109">Entities</script> relate to <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-110-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-110">Things</script>, we
-  require  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-111-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">⊆</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-111">value(e,a) \subseteq
-  value(thingOf(e),a,t)</script> whenever <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-112-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo stretchy="false">∈</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-112">t \in lifetime(e)</script>.  Intuitively, this means that if we are
-  talking about a <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-113-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-113">Thing</script> indirectly by describing an <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-114-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-114">Entity</script>, then
-  any attributes we ascribe to the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-115-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-115">Entity</script> must also describe the
-  associated <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-116-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-116">Thing</script> during their common lifetime.  Attributes of both
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-117-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-117">Entities</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-118-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-118">Things</script> are multi-valued, so there is no
+  In the above description of how <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-103-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-103">Entities</script> relate to <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-104-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-104">Things</script>, we
+  require  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-105-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">⊆</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-105">value(e,a) \subseteq
+  value(thingOf(e),a,t)</script> whenever <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-106-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo stretchy="false">∈</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-106">t \in lifetime(e)</script>.  Intuitively, this means that if we are
+  talking about a <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-107-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-107">Thing</script> indirectly by describing an <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-108-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-108">Entity</script>, then
+  any attributes we ascribe to the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-109-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-109">Entity</script> must also describe the
+  associated <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-110-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-110">Thing</script> during their common lifetime.  Attributes of both
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-111-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-111">Entities</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-112-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><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-112">Things</script> are multi-valued, so there is no
   inconsistency in saying that an entity has two different values for
   some attribute.  In some
   situations, further uniqueness constraints or range constraints
   could be imposed on attributes.
   </p>
-  <p>Only <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-119-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-119">Entities</script> are associated with <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-120-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-120">Things</script>, and this
+  <p>Only <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-113-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-113">Entities</script> are associated with <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-114-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-114">Things</script>, and this
   association is
-  necessary to provide an interpretation for the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-121-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-121">alternateOf</script> and
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-122-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-122">specializationOf</script> relations.  It might also make sense
-  to associate <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-123-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-123">Agents</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-124-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-124">Activities</script>, and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-125-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-125">Interactions</script> with
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-126-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-126">Things</script>, or with some other structures; however, this is not
+  necessary to provide an interpretation for the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-115-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-115">alternateOf</script> and
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-116-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-116">specializationOf</script> relations.  It might also make sense
+  to associate <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-117-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-117">Agents</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-118-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-118">Activities</script>, and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-119-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mi>n</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-119">Interactions</script> with
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-120-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-120">Things</script>, or with some other structures; however, this is not
   necessary to model any of the current features of PROV, so in the
   interest of simplicity we do not do this.
   </p>
@@ -1485,7 +1485,7 @@
 <p>We identify a specific subset of the entities called
   <em>plans</em>:</p>
 <div class="component" id="plans" data-count="4" data-title="Component 4 (plans)"><div class="ruleTitle"><a class="internalDFN" href="#plans">Component 4 (plans)</a></div>
- <p> A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-127-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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-127">Plans \subseteq Entities</script> of plans.</p>
+ <p> A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-121-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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-121">Plans \subseteq Entities</script> of plans.</p>
   </div>
 </section>
 
@@ -1494,8 +1494,8 @@
   <p>We identify another specific subset of the entities called
   <em>collections</em>, with the following associated structure:</p>
   <div class="component" id="collections" data-count="5" data-title="Component 5 (collections)"><div class="ruleTitle"><a class="internalDFN" href="#collections">Component 5 (collections)</a></div>
-    <ul><li>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-128-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-128">Collections \subseteq Entities</script></li>
-    <li>A membership relation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-129-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>O</mi><mi>f</mi><mo stretchy="false">⊆</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-129">MemberOf\subseteq Entities \times Collections</script>
+    <ul><li>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-122-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-122">Collections \subseteq Entities</script></li>
+    <li>A membership relation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-123-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>O</mi><mi>f</mi><mo stretchy="false">⊆</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-123">MemberOf\subseteq Entities \times Collections</script>
   indicating when an entity is a member of another (collection)
   entity.</li>
   </ul>
@@ -1510,10 +1510,10 @@
 <p>An <em>activity</em> is an object that encompasses a set of events.  We introduce:
 </p>
 <div class="component" id="activities" data-count="6" data-title="Component 6 (activities)"><div class="ruleTitle"><a class="internalDFN" href="#activities">Component 6 (activities)</a></div>
-  <ol><li>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-130-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><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-130">Activities \subseteq Objects</script> of activities.</li>
-  <li>Functions <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-131-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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>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-131">startTime : Activities \to Times</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-132-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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>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-132">endTime
+  <ol><li>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-124-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><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-124">Activities \subseteq Objects</script> of activities.</li>
+  <li>Functions <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-125-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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>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-125">startTime : Activities \to Times</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-126-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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>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-126">endTime
   :Activities \to Times</script> giving the start and end time of each activity.</li>
-  <li> Activities are disjoint from Entities: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-133-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">∩</mo><mi>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 mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-133">Entities\cap Activities
+  <li> Activities are disjoint from Entities: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-127-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">∩</mo><mi>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 mathvariant="normal">∅</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-127">Entities\cap Activities
   = \emptyset</script>.</li>
   </ol>
 </div></section>
@@ -1530,7 +1530,7 @@
   entities and activities are disjoint.  We introduce:
 </p>
 <div class="component" id="agents" data-count="7" data-title="Component 7 (agents)"><div class="ruleTitle"><a class="internalDFN" href="#agents">Component 7 (agents)</a></div>
-  <p>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-134-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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-134">Agents \subseteq Objects</script> of agents.</p>
+  <p>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-128-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-128">Agents \subseteq Objects</script> of agents.</p>
   </div>
   <div class="remark">
     <p>There is no requirement that every agent is either an activity
@@ -1541,14 +1541,14 @@
 <section id="influences-1">
 <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
+<p>We consider a set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-129-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-129">Influences \subseteq Objects</script> which has disjoint
   subsets
-  <em>Events</em> connecting entities and activities,
-  <em>Associations</em> between agents and activities,
-    <em>Attributions</em> between entities and agents,
-  <em>Communications</em> between pairs of activities,
-  <em>Delegations</em> between pairs of agents, and
-  <em>Derivations</em> that describe chains of generation and usage
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-130-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-130">Events</script> connecting entities and activities,
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-131-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-131">Associations</script> between agents and activities,
+    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-132-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-132">Attributions</script> between entities and agents,
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-133-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-133">Communications</script> between pairs of activities,
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-134-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-134">Delegations</script> between pairs of agents, and
+  <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>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-135">Derivations</script> that describe chains of generation and usage
   steps.  These kinds of influences are discussed further below.  Influences are disjoint from entities, activities and agents.
 </p>
 <div class="component" id="influences" data-count="8" data-title="Component 8 (influences)"><div class="ruleTitle"><a class="internalDFN" href="#influences">Component 8 (influences)</a></div>
@@ -1569,30 +1569,30 @@
 <section id="events-1">
 <h5><span class="secno">3.2.4.1 </span> Events </h5>
 
-<p>An <em>Event</em> is an influence whose lifetime is a single time
+<p>An <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-144-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-144">Event</script> is an influence whose lifetime is a single time
 instant, and relates an activity to an entity (which could be an
 agent).  Events have types including usage, generation, invalidation, starting and ending.  Events are instantaneous.  We introduce:
 </p>
 <div class="component" id="events" data-count="9" data-title="Component 9 (events)"><div class="ruleTitle"><a class="internalDFN" href="#events">Component 9 (events)</a></div>
-<ol><li> A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-144-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-144">Events \subseteq Influences</script> of events, partitioned
-  into disjoint subsets <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-145-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>s</mi><mo stretchy="false">,</mo><mi>E</mi><mi>n</mi><mi>d</mi><mi>s</mi><mo stretchy="false">,</mo><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">,</mo><mi>U</mi><mi>s</mi><mi>a</mi><mi>g</mi><mi>e</mi><mi>s</mi><mo stretchy="false">,</mo><mi>I</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-145">Starts, Ends, Generations, Usages,
+<ol><li> A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-145-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-145">Events \subseteq Influences</script> of events, partitioned
+  into disjoint subsets <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-146-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>s</mi><mo stretchy="false">,</mo><mi>E</mi><mi>n</mi><mi>d</mi><mi>s</mi><mo stretchy="false">,</mo><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">,</mo><mi>U</mi><mi>s</mi><mi>a</mi><mi>g</mi><mi>e</mi><mi>s</mi><mo stretchy="false">,</mo><mi>I</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-146">Starts, Ends, Generations, Usages,
   Invalidations</script>.
-</li><li> A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-146-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">:</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">→</mo><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-146">time : Events \to Times</script> giving the time of each
-event, such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-147-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-147">lifetime(evt) = \{time(evt)\}</script>.
+</li><li> A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-147-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">:</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">→</mo><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-147">time : Events \to Times</script> giving the time of each
+event, such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-148-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-148">lifetime(evt) = \{time(evt)\}</script>.
 </li>
-<li> A quasi-ordering on events <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-148-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">⪯⊂</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">×</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-148">\preceq \subset Events \times
-Events</script>.  We write <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-149-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo stretchy="false">≺</mo><msup><mi>e</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-149">e \prec e'</script> when <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-150-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo stretchy="false">⪯</mo><msup><mi>e</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-150">e \preceq e'</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-151-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">⪯̸</mo><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-151">e'
+<li> A quasi-ordering on events <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-149-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">⪯⊂</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">×</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-149">\preceq \subset Events \times
+Events</script>.  We write <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-150-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo stretchy="false">≺</mo><msup><mi>e</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-150">e \prec e'</script> when <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-151-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo stretchy="false">⪯</mo><msup><mi>e</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-151">e \preceq e'</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-152-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><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-152">e'
 \not\preceq e</script> hold.
 </li>
-<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-152-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">:</mo><mi>S</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>s</mi><mo stretchy="false">→</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-152">started : Starts \to Activities \times Entities \times Activities</script>.
-</li>
-<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-153-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">:</mo><mi>E</mi><mi>n</mi><mi>d</mi><mi>s</mi><mo stretchy="false">→</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-153">ended : Ends \to Activities \times Entities \times Activities</script>.
+<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-153-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-153">started : Starts \to Activities \times Entities \times Activities</script>.
 </li>
-<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-154-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">:</mo><mi>U</mi><mi>s</mi><mi>a</mi><mi>g</mi><mi>e</mi><mi>s</mi><mo stretchy="false">→</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-154">used : Usages \to Activities \times Entities</script>.
+<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-154-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-154">ended : Ends \to Activities \times Entities \times Activities</script>.
 </li>
-<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-155-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">:</mo><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-155">generated : Generations \to Entities \times Activities</script>.
+<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-155">used : Usages \to Activities \times Entities</script>.
 </li>
-<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-156-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">:</mo><mi>I</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-156">invalidated : Invalidations \to Entities \times Activities</script>.
+<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-156">generated : Generations \to Entities \times Activities</script>.
+</li>
+<li>A function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-157-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-157">invalidated : Invalidations \to Entities \times Activities</script>.
 </li>
 </ol>
 </div>
@@ -1601,11 +1601,12 @@
 
 <h5><span class="secno">3.2.4.2 </span> Associations </h5>
 
-<p>An <em>Association</em> is an influence relating an agent to an activity.  To model associations, we introduce:
+<p>An <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-158-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-158">Association</script> is an influence relating an agent to an activity
+and optional plan.  To model associations, we introduce:
 </p>
 <div class="component" id="associations" data-count="10" data-title="Component 10 (associations)"><div class="ruleTitle"><a class="internalDFN" href="#associations">Component 10 (associations)</a></div>
-  <p>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-157-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-157">Associations \subseteq Influences</script> with associated
-  function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-158-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">:</mo><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>A</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>P</mi><mi>l</mi><mi>a</mi><mi>n</mi><msub><mi>s</mi><mi mathvariant="normal">⊥</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-158">associatedWith : Associations \to  Agents \times Activities \times Plans_\bot</script>.
+  <p>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-159-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>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-159">Associations \subseteq Influences</script> with associated
+  function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-160-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>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-160">associatedWith : Associations \to  Agents \times Activities \times Plans_\bot</script>.
 </p>
   </div>
   </section>
@@ -1613,23 +1614,23 @@
 
 <h5><span class="secno">3.2.4.3 </span> Attributions </h5>
 
-<p>An <em>Attribution</em> is an influence relating an entity to an agent.  To model associations, we introduce:
+<p>An <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>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-161">Attribution</script> is an influence relating an entity to an agent.  To model attributions, we introduce:
 </p>
 <div class="component" id="attributions" data-count="11" data-title="Component 11 (attributions)"><div class="ruleTitle"><a class="internalDFN" href="#attributions">Component 11 (attributions)</a></div>
-  <p>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-159-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-159">Attributions \subseteq Influences</script> with associated
-  function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-160-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>T</mi><mi>o</mi><mo stretchy="false">:</mo><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-160">attributedTo : Attributions \to Entities \times Agents</script>.
+  <p>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-162">Attributions \subseteq Influences</script> with associated
+  function <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>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-163">attributedTo : Attributions \to Entities \times Agents</script>.
 </p>
   </div>
   
 </section>
   <section id="communications-1">
   <h5><span class="secno">3.2.4.4 </span>Communications</h5>
-  <p>A <em>Communication</em> is an influence indicating exchange of
+  <p>A <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>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-164">Communication</script> is an influence indicating exchange of
   information between activities.  To model communications, we introduce:
 </p>
 <div class="component" id="communications" data-count="12" data-title="Component 12 (communications)"><div class="ruleTitle"><a class="internalDFN" href="#communications">Component 12 (communications)</a></div>
-  <p>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-161-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-161">Communications \subseteq Influences</script> with associated
-  function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-162-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">:</mo><mi>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-162">communicatedBy : Communications \to Activities \times Activities</script>.
+  <p>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-165">Communications \subseteq Influences</script> with associated
+  function <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>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-166">communicated : Communications \to Activities \times Activities</script>.
 </p>
   </div>
   
@@ -1637,12 +1638,12 @@
 </section>
   <section id="delegations-1">
   <h5><span class="secno">3.2.4.5 </span>Delegations</h5>
-<p>A <em>Delegation</em> is an influence relating  two agents.  To
+<p>A <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>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-167">Delegation</script> is an influence relating  two agents.  To
   model delegations, we introduce:
 </p>
 <div class="component" id="delegations" data-count="13" data-title="Component 13 (delegations)"><div class="ruleTitle"><a class="internalDFN" href="#delegations">Component 13 (delegations)</a></div>
-  <p>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-163-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-163">Delegations \subseteq Influences</script> and associated function
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-164-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>c</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>o</mi><mi>r</mi><mo stretchy="false">:</mo><mi>D</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>A</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">×</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-164">actedFor : Delegations \to Agents \times Agents \times Activities</script>
+  <p>A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-168">Delegations \subseteq Influences</script> and associated function
+  <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>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-169">actedFor : Delegations \to Agents \times Agents \times Activities</script>
 </p>
   </div>
   
@@ -1651,44 +1652,44 @@
   
   <h5><span class="secno">3.2.4.6 </span> Derivations </h5>
 
-<p>A <em>Derivation</em> is an influence chaining one or more
+<p>A <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>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-170">Derivation</script> is an influence chaining one or more
   generation and use steps.  To model derivations, we introduce an
   auxiliary notion of <em>derivation path</em>.  These paths are of the form </p>
 
-<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-165-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>e</mi><mi>n</mi><msub><mi>t</mi><mi>n</mi></msub><mo stretchy="false">⋅</mo><msub><mi>g</mi><mi>n</mi></msub><mo stretchy="false">⋅</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mi>n</mi></msub><mo stretchy="false">⋅</mo><msub><mi>u</mi><mi>n</mi></msub><mo stretchy="false">⋅</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mrow class="MJX-TeXAtom-ORD"><mi>n</mi><mo stretchy="false">−</mo><mn>1</mn></mrow></msub><mo stretchy="false">⋅</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">⋅</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">⋅</mo><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">⋅</mo><mi>a</mi><mi>c</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">⋅</mo><msub><mi>u</mi><mn>1</mn></msub><mo stretchy="false">⋅</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>0</mn></msub></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-165">ent_n\cdot g_n\cdot  act_n\cdot  u_n\cdot  ent_{n-1}\cdot  ...\cdot
+<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-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" 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-171">ent_n\cdot g_n\cdot  act_n\cdot  u_n\cdot  ent_{n-1}\cdot  ...\cdot
 ent_1\cdot  g_1\cdot  act_1\cdot  u_1\cdot  ent_0</script>
 
-<p>where the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-166-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><msub><mi>t</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-166">ent_i</script> are entities, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-167-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>c</mi><msub><mi>t</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-167">act_i</script> are activities, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-168-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>g</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-168">g_i</script> are generations, and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-169-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-169">u_i</script> are usages.
+<p>where the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>e</mi><mi>n</mi><msub><mi>t</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-172">ent_i</script> are entities, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-173-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>c</mi><msub><mi>t</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-173">act_i</script> are activities, <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"><msub><mi>g</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-174">g_i</script> are generations, and <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"><msub><mi>u</mi><mi>i</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-175">u_i</script> are usages.
 </p>
 <p>Formally, we consider the (regular) language:
 </p>
-<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-170-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mi>s</mi><mo stretchy="false">=</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">⋅</mo><mo stretchy="false">(</mo><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⋅</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">⋅</mo><mi>U</mi><mi>s</mi><mi>a</mi><mi>g</mi><mi>e</mi><mi>s</mi><mo stretchy="false">⋅</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><msup><mo stretchy="false">)</mo><mo stretchy="false">+</mo></msup></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-170">DerivationPaths = Entities \cdot (Generations \cdot Activities \cdot
+<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-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>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-176">DerivationPaths = Entities \cdot (Generations \cdot Activities \cdot
 Usages \cdot Entities)^+</script>
 <p>with the constraints that for each derivation path:
 </p>
 <ul>
-<li>for each substring <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-171-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">⋅</mo><mi>g</mi><mo stretchy="false">⋅</mo><mi>a</mi><mi>c</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-171">ent\cdot g \cdot act</script> we have <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-172-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-172">generated(g) = (ent,act)</script>, and
+<li>for each substring <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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><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-177">ent\cdot g \cdot act</script> we have <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>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-178">generated(g) = (ent,act)</script>, and
 </li>
-<li>for each substring <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-173-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">⋅</mo><mi>u</mi><mo stretchy="false">⋅</mo><mi>e</mi><mi>n</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-173">act \cdot u \cdot ent</script> we have <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-174-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-174">used(u) = (act,ent)</script>.
+<li>for each substring <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-179">act \cdot u \cdot ent</script> we have <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>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-180">used(u) = (act,ent)</script>.
 </li>
 </ul>
 
 
 <div class="component" id="derivations" data-count="14" data-title="Component 14 (derivations)"><div class="ruleTitle"><a class="internalDFN" href="#derivations">Component 14 (derivations)</a></div>
-<p>  A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-175-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">⊆</mo><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-175">Derivations \subseteq Influences</script> with an associated
-  function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-176-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mo stretchy="false">:</mo><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">→</mo><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-176">derivationPath : Derivations \to
+<p>  A set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-181-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><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-181">Derivations \subseteq Influences</script> with an associated
+  function <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>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-182">derivationPath : Derivations \to
 DerivationPaths</script>  linking each derivation to a derivation path.  </p>
 <p></p>
 </div>
 
 <div class="remark">
   <p>
-  The <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-177-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-177">derivationPath</script> function links each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-178-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo stretchy="false">∈</mo><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-178"> d \in Derivations</script> to a
+  The <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-183">derivationPath</script> function links each <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>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-184"> d \in Derivations</script> to a
   derivation path.  A derivation has exactly one associated derivation
   path.  However, if the PROV-N statement <span class="name">wasDerivedFrom(e_2,e_1,-,-,-)</span> is asserted in an
-  instance, there may be multiple derivation paths linking <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-179-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-179">e_2</script> to
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-180-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-180">e_1</script>, each corresponding to a different path, identified by different
-  derivations <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-181-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo stretchy="false">∈</mo><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-181">d \in Derivations</script>.
+  instance, there may be multiple derivation paths linking <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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"><msub><mi>e</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-185">e_2</script> to
+  <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"><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-186">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-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><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-187">d \in Derivations</script>.
   </p>
 
   <p>A derivation path implies the existence of at least one chained generation
@@ -1701,7 +1702,7 @@
   </p>
 <p>
   The reason why we need paths and not just individual derivation
-  steps is to reflect that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-182-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mi>a</mi><mi>s</mi><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>r</mi><mi>o</mi><mi>m</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-182">wasDerivedFrom(id,e_2,e_1,-,-,-,attrs)</script> formulas can
+  steps is to reflect that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-188">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
@@ -1722,131 +1723,195 @@
     
   <ol>
     <li id="axiom1">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-183-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-183">generated(g) = (e,a_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-184-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-184">used(u) = (a_2,e)</script> then there
-    exists <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-185-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo stretchy="false">∈</mo><mi>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-185">c \in Communications</script> such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-186-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-186">communicatedBy(c) = (a_2,a_1)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-189-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-189">generated(g) = (e,a_1)</script> 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"><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-190">used(u) = (a_2,e)</script> then there
+    exists <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-191-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-191">c \in Communications</script> such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-192-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-192">communicated(c) = (a_2,a_1)</script>.
     </li>
     <li id="axiom2">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-187-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-187">started(start) = (a_2,e,a_1)</script> then there exists <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-188-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-188">gen</script> such
-    that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-189-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-189">generated(gen) = (e,a_1)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-193">started(start) = (a_2,e,a_1)</script> then there exists <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></math></span></span></span><script type="math/tex" id="MathJax-Element-194">gen</script> such
+    that <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>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-195">generated(gen) = (e,a_1)</script>.
     </li>
     <li id="axiom3">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-190-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-190">ended(end) = (a_2,e,a_1)</script> then there exists <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-191-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-191">gen</script> such
-    that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-192-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-192">generated(gen) = (e,a_1)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-196">ended(end) = (a_2,e,a_1)</script> then there exists <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>g</mi><mi>e</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-197">gen</script> such
+    that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-198-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><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-198">generated(gen) = (e,a_1)</script>.
     </li>
     <li id="axiom4">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-193-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo stretchy="false">∈</mo><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-193">d \in Derivations</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-194-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mi>r</mi><mi>o</mi><mi>v</mi><mo stretchy="false">:</mo><mi>R</mi><mi>e</mi><mi>v</mi><mi>i</mi><mi>s</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">∈</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">,</mo><mi>p</mi><mi>r</mi><mi>o</mi><mi>v</mi><mo stretchy="false">:</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-194">prov:Revision \in
-    value(d,prov:type)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-195-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">⋅</mo><mi>w</mi><mo stretchy="false">⋅</mo><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-195">derivationPath(deriv) = e_2 \cdot w \cdot
-    e_1</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-196-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-196">thingOf(e_1) = thingOf(e_2)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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> and <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>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-200">prov:Revision \in
+    value(d,prov:type)</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>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-201">derivationPath(deriv) = e_2 \cdot w \cdot
+    e_1</script> then <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>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-202">thingOf(e_1) = thingOf(e_2)</script>.
     </li>
     <li id="axiom5">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-197-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>T</mi><mi>o</mi><mo stretchy="false">(</mo><mi>a</mi><mi>t</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-197">attributedTo(att) = (e,ag)</script> then there exist <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-198-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-198">gen</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-199-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-199">assoc</script>
-    such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-200-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-200">generated(gen) = (e,a)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-201-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-201">associatedWith(assoc) = (a,ag)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-203-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>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-203">attributedTo(att) = (e,ag)</script> then there exist <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></math></span></span></span><script type="math/tex" id="MathJax-Element-204">gen</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-205-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-205">assoc</script>
+    such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-206-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>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-206">generated(gen) = (e,a)</script> and <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>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-207">associatedWith(assoc) = (a,ag)</script>.
     </li>
     <li id="axiom6">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-202-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>c</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>o</mi><mi>r</mi><mo stretchy="false">(</mo><mi>d</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-202">actedFor(deleg) = (ag_2,ag_1,act)</script> then there exist
-    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-203-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><msub><mi>c</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><msub><mi>c</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-203">assoc_1,assoc_2,pl_1,pl_2</script> such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-204-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><msub><mi>c</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-204">associatedWith(assoc_1) = (act,ag_1,pl_1)</script>
-    and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-205-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><msub><mi>c</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-205">associatedWith(assoc_2) = (act,ag_2,pl_2)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-208">actedFor(deleg) = (ag_2,ag_1,act)</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>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-209">assoc_1,assoc_2,pl_1,pl_2</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>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-210">associatedWith(assoc_1) = (ag_1,act,pl_1)</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>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-211">associatedWith(assoc_2) = (ag_2,act,pl_2)</script>.
     </li>
     <li id="axiom7">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-206-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-206">generated(id) = (e,a)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-207-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-207">influenced(id) = (e,a)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-212">generated(id) = (e,a)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-213-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><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-213">influenced(id) = (e,a)</script>.
     </li>
     <li id="axiom8">
-        If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-208-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-208">used(id) = (e,a)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-209-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-209">influenced(id) = (e,a)</script>.
+        If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-214">used(id) = (e,a)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-215-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><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">influenced(id) = (e,a)</script>.
     </li>
     <li id="axiom9">
-            If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-210-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-210">communicatedBy(id) = (a_2,a_1)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-211-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-211">influenced(id) = (a_2,a_1)</script>.
+            If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-216">communicated(id) = (a_2,a_1)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-217-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><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-217">influenced(id) = (a_2,a_1)</script>.
     </li>
     <li id="axiom10">
-     If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-212-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-212">started(id) = (a_2,e,a_1)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-213-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-213">influenced(id) = (a_2,e)</script>.
+     If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-218">started(id) = (a_2,e,a_1)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-219-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>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-219">influenced(id) = (a_2,e)</script>.
     </li>
     <li id="axiom11">
-         If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-214-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-214">ended(id) = (a_2,e,a_1)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-215-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-215">influenced(id) = (a_2,e)</script>.
+         If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-220">ended(id) = (a_2,e,a_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>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-221">influenced(id) = (a_2,e)</script>.
     </li>
     <li id="axiom12">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-216-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-216">invalidated(id) = (e,a)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-217-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-217">influenced(id) = (e,a)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-222">invalidated(id) = (e,a)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-223-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>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">influenced(id) = (e,a)</script>.
     </li>
     <li id="axiom13">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-218-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">⋅</mo><mi>w</mi><mo stretchy="false">⋅</mo><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-218">derivationPath(id) = e_2 \cdot w \cdot e_1</script> then
-    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-219-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-219">influenced(id) = (e_2,e_1)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-224">derivationPath(id) = e_2 \cdot w \cdot e_1</script> then
+    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-225-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><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-225">influenced(id) = (e_2,e_1)</script>.
     </li>
     <li id="axiom14">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-220-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>T</mi><mi>o</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-220">attributedTo(id) = (e,ag)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-221-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-221">influenced(id) = (e,ag)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-226">attributedTo(id) = (e,ag)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-227-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-227">influenced(id) = (e,ag)</script>.
     </li>
     <li id="axiom15">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-222-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-222">associatedWith(id) = (a,ag,pl)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-223-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-223">influenced(id) = (a,ag)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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><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-228">associatedWith(id) = (a,ag,pl)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-229-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>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-229">influenced(id) = (a,ag)</script>.
     </li>
     <li id="axiom16">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-224-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>c</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>o</mi><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-224">actedFor(id) = (ag_2,ag_1)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-225-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-225">influenced(id) = (ag_2,ag_1)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-230">actedFor(id) = (ag_2,ag_1)</script> then <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>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-231">influenced(id) = (ag_2,ag_1)</script>.
     </li>
     <li id="axiom17">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-226-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><msup><mi>n</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-226">generate(gen) = (e,a) = generated(gen')</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-227-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">=</mo><mi>g</mi><mi>e</mi><msup><mi>n</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-227">gen = gen'</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-232">generated(gen) = (e,a) = generated(gen')</script> then <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>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-233">gen = gen'</script>.
     </li>
     <li id="axiom18">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-228-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>n</mi><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>i</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>n</mi><msup><mi>v</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-228">invalidated(inv) = (e,a) = invalidated(inv')</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-229-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>v</mi><mo stretchy="false">=</mo><mi>i</mi><mi>n</mi><msup><mi>v</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-229">inv=inv'</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-234">invalidated(inv) = (e,a) = invalidated(inv')</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-235-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-235">inv=inv'</script>.
     </li>
     <li id="axiom19">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-230-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>s</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-230">started(st) = (a,e_1,a')</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-231-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>s</mi><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-231">started(st') = (a,e_2,a')</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-232-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mo stretchy="false">=</mo><mi>s</mi><msup><mi>t</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-232">st=st'</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-236-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>s</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><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-236">started(st) = (a,e_1,a')</script> and <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>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-237">started(st') = (a,e_2,a')</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>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-238">st=st'</script>.
     </li>
     <li id="axiom20">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-233-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-233">ended(end) = (a,e_1,a')</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-234-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msup><mi>d</mi><mo>′</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-234">ended(end') = (a,e_2,a')</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-235-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">=</mo><mi>e</mi><mi>n</mi><msup><mi>d</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-235">end=end'</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-239-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><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-239">ended(end) = (a,e_1,a')</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-240-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>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-240">ended(end') = (a,e_2,a')</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-241-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>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-241">end=end'</script>.
     </li>
     <li id="axiom21">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-236-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>s</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-236">started(st) = (a,e)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-237-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>t</mi><mo stretchy="false">⪯</mo><mi>e</mi><mi>v</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-237">st \preceq evt</script> for all <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-238-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-238">evt \in events(a)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-242">started(st) = (a,e)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-243-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>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-243">st \preceq evt</script> for all <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-244-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>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-244">evt \in
+    events(a) - Invalidations</script>.
     </li>
     <li id="axiom22">
-        If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-239-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-239">ended(end) = (a,e,a') </script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-240-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">⪯</mo><mi>e</mi><mi>n</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-240">evt \preceq end</script> for all <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-241-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-241">evt \in events(a)</script>.
+        If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-245">ended(end) = (a,e,a') </script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-246-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">⪯</mo><mi>e</mi><mi>n</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-246">evt \preceq end</script> for all
+    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-247-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>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-247">evt \in events(a) - Invalidations</script>.
     </li>
     <li id="axiom23">
-    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-242-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-242">generated(gen) = (e,a)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-243-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">⪯</mo><mi>e</mi><mi>v</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-243">gen \preceq evt</script> for all <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-244-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-244">evt \in events(e)</script>.
+    If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-248">generated(gen) = (e,a)</script> then <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>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-249">gen \preceq evt</script> for all <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>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-250">evt \in events(e)</script>.
     </li>
     <li id="axiom24">
-        If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-245-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>n</mi><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-245">invalidated(inv) = (e,a)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-246-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">⪯</mo><mi>i</mi><mi>n</mi><mi>v</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-246">evt\preceq inv</script> for all <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-247-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-247">evt \in events(e)</script>.
+        If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-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>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-251">invalidated(inv) = (e,a)</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>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-252">evt\preceq inv</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>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-253">evt \in events(e)</script>.
     </li>
     <li id="axiom25">
-    For any derivation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-248-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-248">deriv</script>, with path <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-249-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>w</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-249">derivationPath(deriv) = w</script>,
-    if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-250-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">⋅</mo><mi>g</mi><mo stretchy="false">⋅</mo><mi>a</mi><mo stretchy="false">⋅</mo><mi>u</mi><mo stretchy="false">⋅</mo><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-250">e_2 \cdot g \cdot a \cdot u \cdot e_1 </script> is a substring of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-251-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-251">w</script>
-    where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-252-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">∈</mo><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-252">e_1,e_2 \in Entities</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-253-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo stretchy="false">∈</mo><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-253">g \in Generations</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-254-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo stretchy="false">∈</mo><mi>U</mi><mi>s</mi><mi>a</mi><mi>g</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-254">u \in Usages</script>
-    and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-255-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo stretchy="false">∈</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-255">a \in Activities</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-256-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo stretchy="false">⪯</mo><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-256">u \preceq g</script>.
+    For any derivation <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>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-254">deriv</script>, with path <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>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-255">derivationPath(deriv) = w</script>,
+    if <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"><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-256">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-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>w</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-257">w</script>
+    where <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"><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-258">e_1,e_2 \in Entities</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-259-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><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-259">g \in Generations</script>, <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>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-260">u \in Usages</script>
+    and <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>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-261">a \in Activities</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>u</mi><mo stretchy="false">⪯</mo><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-262">u \preceq g</script>.
     </li>
     <li id="axiom26">
-    For any derivation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-257-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-257">deriv</script>, with path <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-258-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">⋅</mo><mi>w</mi><mo stretchy="false">⋅</mo><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-258">derivationPath(deriv) = e_2
-    \cdot w \cdot e_1</script>, if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-259-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-259">generated(gen_1) = (e_1,a_1)</script> and
-    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-260-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-260">generated(gen_2) = (e_2,a_2)</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-261-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mo stretchy="false">≺</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-261">gen_1 \prec gen_2</script>.  
+    For any derivation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-263-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-263">deriv</script>, with path <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-264-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>P</mi><mi>a</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>d</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><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-264">derivationPath(deriv) = e_2
+    \cdot w \cdot e_1</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"><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-265">generated(gen_1) = (e_1,a_1)</script> and
+    <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>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-266">generated(gen_2) = (e_2,a_2)</script> then <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><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-267">gen_1 \prec gen_2</script>.  
     </li>
+    <li id="axiom27">
+    If  <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>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-268">associatedWith(assoc) = (a,ag,pl)</script>  and <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>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-269">started(start) = (a,e_1,a_1)</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>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-270">invalidated(inv) =
+    (ag,a_2)</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>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-271">start \preceq inv</script>.
+    </li>
+    <li id="axiom28">
+    If  <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>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-272">associatedWith(assoc) = (a,ag,pl)</script>  and <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>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-273">generated(gen) =
+    (ag,a_1)</script> and <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>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-274">ended(end) = (a,e_2,a_2)</script> then <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><mo stretchy="false">⪯</mo><mi>e</mi><mi>n</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-275">gen \preceq end</script>.
+    </li>
+    <li id="axiom29">
+    If  <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>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-276">associatedWith(assoc) = (a,ag,pl)</script>  and <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>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-277">started(start) = (a,e_1,a_1)</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>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-278">ended(end) =
+    (ag,e_2,a_2)</script> then <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>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-279">start \preceq end</script>.
+    </li>
+    <li id="axiom30">
+    If  <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><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-280">associatedWith(assoc) = (a,ag,pl)</script>  and <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>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-281">started(start) =
+    (ag,e_1,a_1)</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>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-282">ended(end) = (a,e_2,a_2)</script> then <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>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-283">start \preceq end</script>.
+    </li>
+       <li id="axiom31">
+    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>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-284">attributedTo(attrib) = (e,ag)</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>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-285">generated(gen_1) =
+    (ag_1,a_1)</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>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-286">generated(gen_2) = (e,a_2)</script> then <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>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-287">gen_1 \preceq gen_2</script>.
+    </li>
+       <li id="axiom32">
+    If <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>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-288">attributedTo(attrib) = (e,ag)</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>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-289">started(start) =
+    (ag_1,e_1,a_1)</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>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-290">generated(gen) = (e,a_2)</script> then <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>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-291">start \preceq gen</script>.
+    </li>
+       <li id="axiom33">
+    If <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>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-292">actedFor(deleg) = (ag_2,ag_1,a)</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>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-293">generated(gen) =
+    (ag_1,a_1)</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>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-294">invalidated(inv) = (ag_2,a_2)</script> then <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><mo stretchy="false">⪯</mo><mi>i</mi><mi>n</mi><mi>v</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-295">gen \preceq inv</script>.
+    </li>
+       <li id="axiom34">
+    If <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>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-296">actedFor(deleg) = (ag_2,ag_1,a)</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>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-297">started(start) =
+    (ag_1,e_1,a_1)</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>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-298">ended(end) = (ag_2,e_2,a_2)</script> then <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>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-299">start \preceq
+    end</script>.
+    </li>
+    
     </ol>
 </div>
 
     <p>These properties are called <em>axioms</em>, and they are
   needed to ensure that the PROV-CONSTRAINTS inferences and
   constraints hold in all structures.</p>
+
+<div class="remark">
+  <p> Axioms 21 and 22 do not require that invalidation events
+  originating from an activity follow the activity's start
+  event(s) or precede its end event(s).
+  This is because
+  there is no such constraint in PROV-CONSTRAINTS.  Arguably, there
+  should be a constraint analogous to Constraint 34 that specifies
+  that any invalidation event in which an activity participates must
+  follow the activity's start event(s) and precede its end event(s).
+  </p></div>
   </section>
 
 <section id="putting-it-all-together">
 <h3><span class="secno">3.4 </span> Putting it all together </h3>
 
-<p>A <em>structure</em> <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>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-262">W</script> is a collection of sets, functions, and relations containing all of the above
-described components.  If we need to talk about the objects or relations of
-more than one structure then we may write <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"><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-263">W_1.Objects</script>, <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"><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-264">W_1.Things</script>,
+<p>A <em>PROV structure</em> <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>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-300">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-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"><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-301">W_1.Objects</script>, <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"><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-302">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>
 
 
-
+<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-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>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-303">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-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>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-304">communicated, associatedWith, attributedTo, actedFor</script> associating
+other types of influences with appropriate data.  On the other hand,
+some features are more distinctive, and represent areas where formal
+modeling has been used to guide the development of PROV.  Derivation
+paths are one such distinctive feature; they correspond to an
+intuition that derivations may describe one or multiple generation-use
+steps leading from one entity to another.  Another distinctive feature
+is the use of <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>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-305">Things</script>, which correspond to changing, real-world
+things, as opposed to <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>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-306">Entities</script>, which correspond to limited views or
+perspectives on <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>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-307">Things</script>, with some fixed aspects.  The semantic
+structures of <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>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-308">Things</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>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-309">Entities</script> provides a foundation for the
+<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>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-310">alternateOf</script> and <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>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-311">specializationOf</script> relations.
+</p>
   
-<div class="note"><div class="note-title"><span>Note</span></div><div class="">
-  TODO: Highlight the distinctive vs obvious/routine features.
-  </div></div>
+
 
 </section>
 <section id="interpretations">
 <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-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 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-265">\rho : Identifiers \to Objects</script> describing
+ An interpretation is a function  <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 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-312">\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-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>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-266">Objects</script> can be denoted by <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>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-267">Identifiers</script>.
+ <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>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-313">Objects</script> can be denoted by <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>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-314">Identifiers</script>.
 </p>
 </section>
 
@@ -1855,59 +1920,59 @@
 <section id="semantics">
 <!--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-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>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-268">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-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 mathvariant="italic">ρ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-269">\rho</script> be an interpretation of identifiers as objects in <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>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-270">W</script>.
+<p>In what follows, let <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>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-315">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-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 mathvariant="italic">ρ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-316">\rho</script> be an interpretation of identifiers as objects in <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>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-317">W</script>.
 The annotations [WF] refer to well-formedness constraints that correspond to typing constraints.
 </p>
 
 <section id="satisfaction">
 <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-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 mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-271">\phi</script>, a structure <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>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-272">W</script> and an interpretation
- <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 mathvariant="italic">ρ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-273">\rho</script>.
-We define notation <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>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-274">W,\rho \models \phi</script> which means that <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 mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-275">\phi</script> is
- satisfied in <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><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-276">W,\rho</script>. For atomic formulas, the definition of the
+<p>Consider a formula <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 mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-318">\phi</script>, a structure <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> and an interpretation
+ <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 mathvariant="italic">ρ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-320">\rho</script>.
+We define notation <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>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-321">W,\rho \models \phi</script> which means that <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 mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-322">\phi</script> is
+ satisfied in <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>W</mi><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-323">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-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>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-277">W,\rho \models True</script> always holds.</li>
-  <li><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>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-278">W,\rho \models False</script> never holds.</li>
-  <li><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>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-279">W,\rho \models x = y</script> holds if and only if <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 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-280">\rho(x) =
+  <li><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><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-324">W,\rho \models True</script> always holds.</li>
+  <li><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>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-325">W,\rho \models False</script> never holds.</li>
+  <li><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><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-326">W,\rho \models x = y</script> holds if and only if <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">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-327">\rho(x) =
     \rho(y)</script>.</li>
-  <li><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>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-281">W,\rho \models \neg \phi</script> holds if and only if <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>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-282">W,\rho \models
+  <li><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><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-328">W,\rho \models \neg \phi</script> holds if and only if <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>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-329">W,\rho \models
   \phi</script> does not hold.</li>
-  <li><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>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-283">W,\rho \models \phi \wedge \psi</script> holds if and only 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>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-284">W,\rho \models
-  \phi</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>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-285">W,\rho \models \psi</script>.</li>
-  <li><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>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-286">W,\rho \models \phi \vee \psi</script> holds if either <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>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-287">W,\rho \models \phi</script>
-  or <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>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-288">W,\rho \models \psi</script>.</li>
-  <li><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>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-289">W,\rho \models \phi \Rightarrow \psi</script> holds if <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>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-290">W,\rho \models \phi</script>
-  implies <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>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-291">W,\rho \models \psi</script>.</li>
-  <li><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>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-292">W,\rho \models \exists x. \phi</script> holds if there exists some <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>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-293">obj \in
-  Objects</script> such that <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>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-294">W,\rho[x:=obj] \models \phi</script>.</li>
-  <li><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>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-295">W,\rho \models \forall x. \phi</script> holds if there for every <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>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-296">obj \in
-  Objects</script> we have <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>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-297">W,\rho[x:=obj] \models \phi</script>.</li>
+  <li><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><mo stretchy="false">∧</mo><mi mathvariant="italic">ψ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-330">W,\rho \models \phi \wedge \psi</script> holds if and only if <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>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-331">W,\rho \models
+  \phi</script> and <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><mo stretchy="false">⊨</mo><mi mathvariant="italic">ψ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-332">W,\rho \models \psi</script>.</li>
+  <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 mathvariant="italic">ϕ</mi><mo stretchy="false">∨</mo><mi mathvariant="italic">ψ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-333">W,\rho \models \phi \vee \psi</script> holds if either <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 mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-334">W,\rho \models \phi</script>
+  or <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 mathvariant="italic">ψ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-335">W,\rho \models \psi</script>.</li>
+  <li><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><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-336">W,\rho \models \phi \Rightarrow \psi</script> holds if <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="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-337">W,\rho \models \phi</script>
+  implies <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">ψ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-338">W,\rho \models \psi</script>.</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="normal">∃</mi><mi>x</mi><mo>.</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-339">W,\rho \models \exists x. \phi</script> holds if there exists some <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>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-340">obj \in
+  Objects</script> such 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>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-341">W,\rho[x:=obj] \models \phi</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="normal">∀</mi><mi>x</mi><mo>.</mo><mi mathvariant="italic">ϕ</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-342">W,\rho \models \forall x. \phi</script> holds if there for every <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>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-343">obj \in
+  Objects</script> we have <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>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-344">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-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>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-298">Objects</script>; that is, quantifiers range over entities,
+  the set of <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>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-345">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-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>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-299">Things</script> and relations
+  subdivided into types of influences).  <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>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-346">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-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>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-300">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-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>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-347">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-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>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-301">I</script> as a set of atomic formulas <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"><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-302">\{\phi_1,\ldots,\phi_n\}</script>, or equivalently a
-    single formula <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 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-303">\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-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"><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-304">x_1,\ldots,x_k</script> are the existential
-    variables of <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>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-305">I</script>.
+    instance <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>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-348">I</script> as a set of atomic formulas <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"><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-349">\{\phi_1,\ldots,\phi_n\}</script>, or equivalently a
+    single formula <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 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-350">\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-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"><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-351">x_1,\ldots,x_k</script> are the existential
+    variables of <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>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-352">I</script>.
     </p>
     </div>
 
@@ -1917,9 +1982,9 @@
 <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-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>o</mi><mi>b</mi><mi>j</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-306">obj</script> matches attributes <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"><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-307">[attr_1=val_1,...]</script> in structure <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>W</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-308">W</script> provided:
-for each attribute <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>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-309">attr_i</script>, we have <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>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-310">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-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>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-311">match(W,obj,attrs)</script>.
+<p>We say that an object <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>o</mi><mi>b</mi><mi>j</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-353">obj</script> matches attributes <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"><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-354">[attr_1=val_1,...]</script> in structure <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></math></span></span></span><script type="math/tex" id="MathJax-Element-355">W</script> provided:
+for each attribute <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>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-356">attr_i</script>, we have <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>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-357">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-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>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-358">match(W,obj,attrs)</script>.
 </p>
     
 </section>
@@ -1931,18 +1996,18 @@
 
 <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-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>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-312">entity(id,attrs)</script> where <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>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-313">id</script> denotes an entity.
+<p>An entity formula is of the form <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>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-359">entity(id,attrs)</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"><mi>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-360">id</script> denotes an entity.
 </p>
-<p>Entity formulas <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>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-314">entity(id,attrs)</script> can be interpreted as follows:
+<p>Entity formulas <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>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-361">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-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>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-315">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-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>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-362">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-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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-316">id</script> denotes an entity <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>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-317">ent = \rho(id) \in Entities</script>
+<li>[WF] <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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-363">id</script> denotes an entity <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>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-364">ent = \rho(id) \in Entities</script>
 </li>
-<li>There exists <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>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-318">gen,a</script> such that <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>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-319">generated(gen) = (e,a)</script>.</li>
-<li>There exists <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>i</mi><mi>n</mi><mi>v</mi><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-320">inv,a'</script> such that <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>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-321">invalidated(inv) = (e,a)</script>.</li>
-<li>the attributes match: <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>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-322">match(W,ent, attrs)</script>.
+<li>There exists <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>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-365">gen,a</script> such that <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>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-366">generated(gen) = (e,a)</script>.</li>
+<li>There exists <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><mi>n</mi><mi>v</mi><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-367">inv,a'</script> such that <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>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-368">invalidated(inv) = (e,a)</script>.</li>
+<li>the attributes match: <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>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-369">match(W,ent, attrs)</script>.
 </li>
 </ol>
 
@@ -1950,7 +2015,7 @@
 </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-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>x</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-323">x</script> denotes an entity <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>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-324">e</script> such that <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>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-325">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-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>x</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-370">x</script> denotes an entity <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>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-371">e</script> such that <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>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-372">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])
@@ -1968,25 +2033,25 @@
 
 <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-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>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-326">activity(id,st,et,attrs)</script>
-where <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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-327">id</script> is a identifier referring to the activity, <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>s</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-328">st</script> is a start
-time and <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>e</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-329">et</script> is an end time, 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>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-330">attrs</script> are the attributes of
-activity <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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-331">id</script>.
+<p>An activity formula  is of the form <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>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-373">activity(id,st,et,attrs)</script>
+where <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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-374">id</script> is a identifier referring to the activity, <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>s</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-375">st</script> is a start
+time and <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><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-376">et</script> is an end time, and <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>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-377">attrs</script> are the attributes of
+activity <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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-378">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-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><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-332">W,\rho \models activity(id,st,et,attrs)</script>
+  <p><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>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-379">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-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></math></span></span></span><script type="math/tex" id="MathJax-Element-333">id</script> maps to an activity <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>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-334">act = \rho(id) \in Activities</script>
-</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 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-335">\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-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>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-336">startTime(id) = \rho(st)</script>
+<li>[WF] The identifier <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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-380">id</script> maps to an activity <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>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-381">act = \rho(id) \in Activities</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 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-337">\rho(et)</script> is the activity's end time, that is: <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>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-338">endTime(id) = \rho(et)</script>
+<li><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 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-382">\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-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>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-383">startTime(id) = \rho(st)</script>
 </li>
-<li>There exists <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>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-339">start,e,a</script> such that <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>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-340">started(start) = (act,e,a)</script>.</li>
-<li>There exists <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>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-341">end,e',a'</script> such that <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>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-342">ended(end) = (act,e',a')</script>.</li>
-<li>The attributes match: <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>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-343">match(W,act,attrs)</script>.
+<li><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 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-384">\rho(et)</script> is the activity's end time, that is: <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>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-385">endTime(id) = \rho(et)</script>
+</li>
+<li>There exists <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>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-386">start,e,a</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>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-387">started(start) = (act,e,a)</script>.</li>
+<li>There exists <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>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-388">end,e',a'</script> such that <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>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-389">ended(end) = (act,e',a')</script>.</li>
+<li>The attributes match: <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>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-390">match(W,act,attrs)</script>.
 </li>
 </ol>
 </div>
@@ -1995,15 +2060,15 @@
 
 <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-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>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-344">agent(id,attrs)</script> where <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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-345">id</script> denotes the agent and <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>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-346">attrs</script> describes additional attributes.
+<p>An agent formula is of the form <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>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-391">agent(id,attrs)</script> where <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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-392">id</script> denotes the agent and <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>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-393">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-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>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-347">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-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>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-394">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-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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-348">id</script> denotes an agent <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>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-349">ag = \rho(id) \in Agents</script>
+    <li>[WF] <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> denotes an agent <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><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-396">ag = \rho(id) \in Agents</script>
     </li>
-    <li>The attributes match: <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>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-350">match(W,ag,attrs)</script>.
+    <li>The attributes match: <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>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-397">match(W,ag,attrs)</script>.
     </li>
   </ol>
 </div>
@@ -2018,25 +2083,25 @@
 <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-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><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-351">wasGeneratedBy(id,e,a,t,attrs)</script> where <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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-352">id</script> is an event identifier,
-<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>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-353">e</script> is an entity identifier, <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>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-354">a</script> is an activity identifier, <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>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-355">attrs</script> is
-a set of attribute-value pairs, and <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>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-356">t</script> is a time.
+<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>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-398">wasGeneratedBy(id,e,a,t,attrs)</script> where <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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-399">id</script> is an event identifier,
+<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></math></span></span></span><script type="math/tex" id="MathJax-Element-400">e</script> is an entity identifier, <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></math></span></span></span><script type="math/tex" id="MathJax-Element-401">a</script> is an activity identifier, <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>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-402">attrs</script> is
+a set of attribute-value pairs, and <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>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-403">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-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>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-357">W,\rho \models
+ <p><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>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-404">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-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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-358">id</script> denotes a generation event <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>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-359">evt = \rho(id) \in Generations</script>.
-</li>
-<li>[WF] The identifier <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>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-360">e</script> denotes an entity <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>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-361">ent = \rho(e) \in Entities</script>.
+<li>[WF] The 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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-405">id</script> denotes a generation event <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>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-406">evt = \rho(id) \in Generations</script>.
 </li>
-<li>[WF] The identifier <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>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-362">a</script> denotes an activity <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>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-363">act = \rho(a) \in Activities</script>.
+<li>[WF] The 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>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-407">e</script> denotes an entity <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>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-408">ent = \rho(e) \in Entities</script>.
 </li>
-<li>The event <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>e</mi><mi>v</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-364">evt</script> occurred at time <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 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-365">\rho(t) \in Times</script>, i.e. <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>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-366">time(evt) = \rho(t)</script>.
+<li>[WF] The identifier <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>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-409">a</script> denotes an activity <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>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-410">act = \rho(a) \in Activities</script>.
 </li>
-<li>The activity <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>a</mi><mi>c</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-367">act</script> generated <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></math></span></span></span><script type="math/tex" id="MathJax-Element-368">ent</script> via <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>e</mi><mi>v</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-369">evt</script>, i.e. <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>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-370">generated(evt) = (ent,act)</script>.
+<li>The 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></math></span></span></span><script type="math/tex" id="MathJax-Element-411">evt</script> occurred at time <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 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-412">\rho(t) \in Times</script>, i.e. <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>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-413">time(evt) = \rho(t)</script>.
 </li>
-<li>The attribute values match: <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>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-371">match(W,evt,attrs)</script>.
+<li>The activity <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><mi>c</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-414">act</script> generated <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><mi>n</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-415">ent</script> via <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>, i.e. <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>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-417">generated(evt) = (ent,act)</script>.
+</li>
+<li>The attribute values match: <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>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-418">match(W,evt,attrs)</script>.
 </li>
 </ol>
 </div>
@@ -2045,49 +2110,49 @@
 <section id="use">
 <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-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>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-372">used(id,a,e,t,attrs)</script> where <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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-373">id</script>
-denotes an event, <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>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-374">a</script> is an activity identifier, <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>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-375">e</script> is an object
-identifier, <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>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-376">attrs</script> is a set of attribute-value pairs, and <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>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-377">t</script> is a time.
+<p>The use formula is of the form <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>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-419">used(id,a,e,t,attrs)</script> where <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 an 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>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-421">a</script> is an activity 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> is an object
+identifier, <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>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-423">attrs</script> is a set of attribute-value pairs, and <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>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-424">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-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>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-378">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-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>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-425">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-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 a usage event <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>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-380">evt = \rho(id) \in Usages</script>.
-</li>
-<li>[WF] The identifier <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>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-381">a</script> denotes an activity <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>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-382">act = \rho(id) \in Activities</script>.
+<li>[WF] The identifier <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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-426">id</script> denotes a usage event <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><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-427">evt = \rho(id) \in Usages</script>.
 </li>
-<li>[WF] The identifier <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></math></span></span></span><script type="math/tex" id="MathJax-Element-383">e</script> denotes an entity <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>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-384">ent = \rho(e) \in Entities</script>.
+<li>[WF] The 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></math></span></span></span><script type="math/tex" id="MathJax-Element-428">a</script> denotes an 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><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-429">act = \rho(id) \in Activities</script>.
 </li>
-<li>The event <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>e</mi><mi>v</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-385">evt</script> occurred at time <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 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-386">\rho(t) \in Times</script>,
-i.e. <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>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-387">time(evt) = \rho(t)</script>.
+<li>[WF] The identifier <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></math></span></span></span><script type="math/tex" id="MathJax-Element-430">e</script> denotes an entity <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>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-431">ent = \rho(e) \in Entities</script>.
 </li>
-<li>The activity <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></math></span></span></span><script type="math/tex" id="MathJax-Element-388">act</script> used <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>o</mi><mi>b</mi><mi>j</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-389">obj</script> via <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>v</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-390">evt</script>, i.e. <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>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-391">used(evt) = (act,ent)</script>.
+<li>The 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></math></span></span></span><script type="math/tex" id="MathJax-Element-432">evt</script> occurred at time <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 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-433">\rho(t) \in Times</script>,
+i.e. <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>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-434">time(evt) = \rho(t)</script>.
 </li>
-<li>The attribute values match: <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>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-392">match(W,evt,attrs)</script>.
+<li>The activity <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>a</mi><mi>c</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-435">act</script> used <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>o</mi><mi>b</mi><mi>j</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-436">obj</script> via <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>, i.e. <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>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-438">used(evt) = (act,ent)</script>.
+</li>
+<li>The attribute values match: <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>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-439">match(W,evt,attrs)</script>.
 </li></ol></div>
 
 </section>
 <section id="invalidation">
 <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-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>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-393">wasInvalidatedBy(id,e,a,t,attrs)</script> where <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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-394">id</script> is an event 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>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-395">e</script> is an entity identifier, <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></math></span></span></span><script type="math/tex" id="MathJax-Element-396">a</script> is an activity identifier, <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>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-397">attrs</script> is a set of attribute-value pairs, 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>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-398">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-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>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-399">W,\rho \models
+<p>The invalidation formula is of the form <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><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-440">wasInvalidatedBy(id,e,a,t,attrs)</script> where <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> is an event identifier, <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></math></span></span></span><script type="math/tex" id="MathJax-Element-442">e</script> is an entity 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> is an activity identifier, <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>t</mi><mi>t</mi><mi>r</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-444">attrs</script> is a set of attribute-value pairs, and <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>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-445">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-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>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-446">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-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 invalidation event <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>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-401">evt = \rho(id) \in Invalidations</script>.
-</li>
-<li>[WF] The identifier <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>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-402">e</script> denotes an entity <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>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-403">ent = \rho(e) \in Entities</script>.
-</li>
-<li>[WF] The identifier <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>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-404">a</script> denotes an activity <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>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-405">act = \rho(a) \in Activities</script>.
+<li>[WF] The 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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-447">id</script> denotes an invalidation event <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>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-448">evt = \rho(id) \in Invalidations</script>.
 </li>
-<li>The event <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>e</mi><mi>v</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-406">evt</script> occurred at time <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 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-407">\rho(t) \in Times</script>,
-  i.e. <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><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-408">time(evt) = \rho(t)</script>.
+<li>[WF] The 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>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-449">e</script> denotes an entity <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>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-450">ent = \rho(e) \in Entities</script>.
 </li>
-<li>The activity <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>a</mi><mi>c</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-409">act</script> invalidated <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>e</mi><mi>n</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-410">ent</script> via <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></math></span></span></span><script type="math/tex" id="MathJax-Element-411">evt</script>, i.e. <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>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-412">invalidated(evt) = (ent,act)</script>.
+<li>[WF] The identifier <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>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-451">a</script> denotes an activity <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>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-452">act = \rho(a) \in Activities</script>.
 </li>
-<li>The attribute values match: <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>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-413">match(W,evt,attrs)</script>.
+<li>The 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></math></span></span></span><script type="math/tex" id="MathJax-Element-453">evt</script> occurred at time <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 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-454">\rho(t) \in Times</script>,
+  i.e. <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>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-455">time(evt) = \rho(t)</script>.
+</li>
+<li>The activity <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><mi>c</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-456">act</script> invalidated <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><mi>n</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-457">ent</script> via <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>, i.e. <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>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-459">invalidated(evt) = (ent,act)</script>.
+</li>
+<li>The attribute values match: <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>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-460">match(W,evt,attrs)</script>.
 </li></ol></div>
 </section>
 
@@ -2095,81 +2160,81 @@
 
 <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-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>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-414">wasAssociatedWith(id,a,ag,pl,attrs)</script>.</p>
+<p>An association formula has the form <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><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-461">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-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>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-415">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-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>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-462">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-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><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-416">assoc</script> denotes an association <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>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-417">assoc = \rho(id) \in Associations</script>.
-</li>
-<li>[WF] <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>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-418">a</script> denotes an 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><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-419">act = \rho(a) \in Activities</script>.
+<li>[WF] <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>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-463">assoc</script> denotes an association <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>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-464">assoc = \rho(id) \in Associations</script>.
 </li>
-<li>[WF] <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>a</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-420">ag</script> denotes an agent <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>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-421">agent = \rho(ag) \in Agents</script>.
+<li>[WF] <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>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-465">a</script> denotes an activity <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><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-466">act = \rho(a) \in Activities</script>.
 </li>
-<li>[WF] <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>p</mi><mi>l</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-422">pl</script> denotes a plan <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>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-423">plan=\rho(pl) \in Plans</script>.
+<li>[WF] <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>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-467">ag</script> denotes an agent <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>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-468">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-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><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-424">associatedWith(assoc) = (agent,act,plan)</script>.
+<li>[WF] <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>p</mi><mi>l</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-469">pl</script> denotes a plan <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>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-470">plan=\rho(pl) \in Plans</script>.
 </li>
-<li>The attributes match: <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>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-425">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-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>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-471">associatedWith(assoc) = (agent,act,plan)</script>.
+</li>
+<li>The attributes match: <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>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-472">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-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>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-426">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-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>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-473">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-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>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-427">assoc</script> denotes an association <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>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-428">assoc = \rho(id) \in Associations</script>.
+<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>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-474">assoc</script> denotes an association <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>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-475">assoc = \rho(id) \in Associations</script>.
 </li>
-<li>[WF] <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></math></span></span></span><script type="math/tex" id="MathJax-Element-429">a</script> denotes an activity <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>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-430">act = \rho(a) \in Activities</script>.
+<li>[WF] <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></math></span></span></span><script type="math/tex" id="MathJax-Element-476">a</script> denotes an activity <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>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-477">act = \rho(a) \in Activities</script>.
 </li>
-<li>[WF] <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>a</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-431">ag</script> denotes an agent <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>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-432">agent = \rho(ag) \in Agents</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>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-478">ag</script> denotes an agent <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>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-479">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-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><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-433">associatedWith(assoc) = (agent,act,\bot)</script>.
+  plan, i.e. <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><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-480">associatedWith(assoc) = (agent,act,\bot)</script>.
 </li>
-<li>The attributes match: <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>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-434">match(W,assoc,attrs)</script>.
+<li>The attributes match: <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>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-481">match(W,assoc,attrs)</script>.
 </li></ol></div>
 </section>
 <section id="start">
 <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-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>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-435">wasStartedBy(id,a_2,e,a_1,t,attrs)</script> is interpreted as follows:</p>
+<p>A start formula <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>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-482">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-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>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-436">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-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>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-483">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-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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-437">id</script> denotes a start event <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>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-438">evt = \rho(id) \in Starts</script>.
-</li>
-<li>[WF] <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"><msub><mi>a</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-439">a_2</script> denotes an 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><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-440">act_2 = \rho(a_2) \in Activities</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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-484">id</script> denotes a start event <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>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-485">evt = \rho(id) \in Starts</script>.
 </li>
-<li>[WF] <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>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-441">e</script> denotes an entity <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>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-442">ent = \rho(e) \in Entities</script>.
-</li>
-<li>[WF] <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"><msub><mi>a</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-443">a_1</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><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-444">act_1 = \rho(a_1) \in Activities</script>.
+<li>[WF] <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"><msub><mi>a</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-486">a_2</script> denotes an activity <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>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-487">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-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>a</mi><mi>c</mi><msub><mi>t</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-445">act_2</script>, that is, <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 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-446">\rho(t) = startTime(act_2) = time(evt)</script>.
+<li>[WF] <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>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-488">e</script> denotes an entity <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>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-489">ent = \rho(e) \in Entities</script>.
 </li>
-<li> The activity <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>a</mi><mi>c</mi><msub><mi>t</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-447">act_1</script> started <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><mi>c</mi><msub><mi>t</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-448">act_2</script> via entity <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>e</mi><mi>n</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-449">ent</script>: that is,
-  <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>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-450">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-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>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-451">match(W,evt,attrs)</script>.
+<li>[WF] <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"><msub><mi>a</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-490">a_1</script> denotes an activity <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><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-491">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-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></math></span></span></span><script type="math/tex" id="MathJax-Element-492">act_2</script>, that is, <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 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-493">\rho(t) = startTime(act_2) = time(evt)</script>.
+</li>
+<li> The activity <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>c</mi><msub><mi>t</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-494">act_1</script> started <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>c</mi><msub><mi>t</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-495">act_2</script> via entity <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>e</mi><mi>n</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-496">ent</script>: that is,
+  <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>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-497">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-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>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-498">match(W,evt,attrs)</script>.
 </li></ol></div>
 </section>
 <section id="end">
 <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-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>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-452">wasEndedBy(id,a_2,e,a_1,t,attrs)</script> is interpreted as follows:</p>
+<p>An activity end formula <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>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-499">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-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>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-453">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-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>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-500">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-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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-454">id</script> denotes an end event <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>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-455">evt = \rho(id) \in Ends</script>.</li>
-<li> [WF] <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"><msub><mi>a</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-456">a_2</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><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-457">act_2 = \rho(a_2)\in Activities</script>.</li>
-<li> [WF] <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></math></span></span></span><script type="math/tex" id="MathJax-Element-458">e</script> denotes an entity <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>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-459">ent = \rho(e)\in Entities</script>.</li>
-<li>[WF] <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"><msub><mi>a</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-460">a_1</script> denotes an 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><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-461">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-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>a</mi><mi>c</mi><msub><mi>t</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-462">act_2</script>, that is, <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 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-463">\rho(t) = endTime(act_2) = time(evt)</script>.</li>
-<li> The activity <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>a</mi><mi>c</mi><msub><mi>t</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-464">act_1</script> ended <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>a</mi><mi>c</mi><msub><mi>t</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-465">act_2</script> via entity <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>e</mi><mi>n</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-466">ent</script>: that is,
-  <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>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-467">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-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>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-468">match(W,evt,attrs)</script>.</li>
+  <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"><mi>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-501">id</script> denotes an end event <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>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-502">evt = \rho(id) \in Ends</script>.</li>
+<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"><msub><mi>a</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-503">a_2</script> denotes an activity <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>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-504">act_2 = \rho(a_2)\in Activities</script>.</li>
+<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"><mi>e</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-505">e</script> denotes an entity <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>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-506">ent = \rho(e)\in Entities</script>.</li>
+<li>[WF] <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"><msub><mi>a</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-507">a_1</script> denotes an activity <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>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-508">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-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></math></span></span></span><script type="math/tex" id="MathJax-Element-509">act_2</script>, that is, <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 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-510">\rho(t) = endTime(act_2) = time(evt)</script>.</li>
+<li> The activity <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>a</mi><mi>c</mi><msub><mi>t</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-511">act_1</script> ended <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>a</mi><mi>c</mi><msub><mi>t</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-512">act_2</script> via entity <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>e</mi><mi>n</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-513">ent</script>: that is,
+  <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>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-514">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-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>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-515">match(W,evt,attrs)</script>.</li>
 </ol>
 </div>
 </section>
@@ -2178,24 +2243,24 @@
 <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-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>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-469">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-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>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-516">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-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>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-470">W,\rho \models wasAttributedTo(id,e,ag,attrs)</script>
+  <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>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-517">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-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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-471">id</script> denotes an association <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>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-472">assoc = \rho(id) \in Associations</script>.
+<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"><mi>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-518">id</script> denotes an association <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>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-519">assoc = \rho(id) \in Associations</script>.
 </li>
-<li>[WF] <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></math></span></span></span><script type="math/tex" id="MathJax-Element-473">e</script> denotes an entity <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>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-474">ent = \rho(e) \in Entities</script>.
+<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-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>a</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-475">ag</script> denotes an agent <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>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-476">agent = \rho(ag) \in Agents</script>.
+<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"><mi>a</mi><mi>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-522">ag</script> denotes an agent <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>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-523">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-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>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-477">attributedTo(assoc)
+<li>The entity was attributed to the agent, i.e. <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>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-524">attributedTo(assoc)
 = (ent,agent)</script>.
 </li>
-<li>The attributes match: <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>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-478">match(W,assoc,attrs)</script>.
+<li>The attributes match: <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>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-525">match(W,assoc,attrs)</script>.
 </li></ol>
 </div>
 </section>
@@ -2203,23 +2268,23 @@
 
 <section id="communication">
 <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-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>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-479">wasInformedBy(id,a_2,a_2,attrs)</script> is
+<p>A communication formula <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>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-526">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-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>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-480">W,\rho \models wasInformedBy(id,a_2,a_1,attrs)</script>
+  <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>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-527">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-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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-481">id</script> denotes a communication <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>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-482">comm = \rho(id) \in Communications</script>.
+<li>[WF] <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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-528">id</script> denotes a communication <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>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-529">comm = \rho(id) \in Communications</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"><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-483">a_1,a_2</script> denote  activities <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>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-484">act_1 = \rho(a_1) \in
+<li>[WF] <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"><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-530">a_1,a_2</script> denote  activities <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>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-531">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-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>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-485">gen,use,ent</script> such that <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>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><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-486">communicatedBy(comm) =
-(act_2,act_1)</script> and <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>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-487">generated(gen) = (ent,act_1)</script> and <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>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-488">used(use) = (act_2,ent)</script>.
+<li>There exist <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>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-532">gen,use,ent</script> such that <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>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-533">communicated(comm) =
+(act_2,act_1)</script> and <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>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-534">generated(gen) = (ent,act_1)</script> and <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>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-535">used(use) = (act_2,ent)</script>.
 </li>
-<li>The attributes match: <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>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-489">match(W,comm,attrs)</script>.
+<li>The attributes match: <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>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-536">match(W,comm,attrs)</script>.
 </li></ol>
 </div>
 </section>
@@ -2229,22 +2294,22 @@
 
 <h4><span class="secno">4.4.9 </span> Delegation </h4>
 
-<p>The <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>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-490">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-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><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-491">ActsFor</script> relation as follows:</p>
+<p>The <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>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-537">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-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>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-538">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-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>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-492">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-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>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-539">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-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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-493">id</script> denotes a delegation <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>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-494">deleg=\rho(id) \in Delegations</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"><mi>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-495">a</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><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-496">act=\rho(a) \in Activities</script>.
+<li>[WF] <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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-540">id</script> denotes a delegation <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>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-541">deleg=\rho(id) \in Delegations</script>.
 </li>
-<li>[WF] <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><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-497">ag_1,ag_2</script> denote agents <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>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-498">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-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></math></span></span></span><script type="math/tex" id="MathJax-Element-542">a</script> denotes an activity <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><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-543">act=\rho(a) \in Activities</script>.
 </li>
-<li>The agent <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>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-499">agent_2</script> acted for the agent <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>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-500">agent_1</script> with respect to
-  the activity <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>a</mi><mi>c</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-501">act</script>, i.e. <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><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-502">actedFor(deleg) = (agent_2,agent_1,act)</script>.
+<li>[WF] <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>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-544">ag_1,ag_2</script> denote agents <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>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-545">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-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>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-503">match(W,deleg,attrs)</script>.
+<li>The agent <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>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-546">agent_2</script> acted for the agent <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><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-547">agent_1</script> with respect to
+  the 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></math></span></span></span><script type="math/tex" id="MathJax-Element-548">act</script>, i.e. <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><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-549">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-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>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-550">match(W,deleg,attrs)</script>.
 </li></ol></div>
 </section>
 
@@ -2255,10 +2320,10 @@
 
 <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-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>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-504">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-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>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-551">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-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><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-505">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-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>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-552">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>
@@ -2266,26 +2331,26 @@
 <section id="precise">
 <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-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>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-506">wasDerivedFrom(id,e_2,e_1,a,g,u,attrs)</script>.
+<p>A precise derivation formula has the form <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>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-553">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-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>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-507">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-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>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-554">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-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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-508">id</script> denotes a derivation <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>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-509">deriv = \rho(id) \in Derivations</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 derivation <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>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-556">deriv = \rho(id) \in Derivations</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"><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-510">e_1,e_2</script> denote entities <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><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-511">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-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"><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-557">e_1,e_2</script> denote entities <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>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-558">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-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>a</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-512">a</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><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-513">act = \rho(a) \in Activities</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></math></span></span></span><script type="math/tex" id="MathJax-Element-559">a</script> denotes an activity <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>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-560">act = \rho(a) \in Activities</script>.
 </li>
-<li>[WF] <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>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-514">g</script> denotes a generation event <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>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-515">gen = \rho(g) \in Generations</script>.
+<li>[WF] <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>g</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-561">g</script> denotes a generation event <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>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-562">gen = \rho(g) \in Generations</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>u</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-516">u</script> denotes a use 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 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-517">\rho(u) \in Usages</script>.
+<li>[WF] <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>u</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-563">u</script> denotes a use event <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 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-564">\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-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>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-518">derivationPath(deriv) =
+entities via the activity, generation and use: <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>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-565">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-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>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-519">match(W,deriv,attrs)</script>.</li>
+<li>The attribute values match: <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>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-566">match(W,deriv,attrs)</script>.</li>
 </ol>
 </div>
 </section>
@@ -2293,18 +2358,18 @@
 
 <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-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>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-520">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-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>.</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-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><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-521">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-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><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-568">W,\rho \models wasDerivedFrom(id,e_2,e_1,-,-,-,attrs)</script> holds if and only if:</p>
 <ol>
-<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"><mi>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-522">id</script> denotes a derivation <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>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-523">deriv = \rho(id) \in Derivations</script>.
+<li>[WF] <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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-569">id</script> denotes a derivation <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><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-570">deriv = \rho(id) \in Derivations</script>.
 </li>
-<li>[WF] <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"><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-524">e_1,e_2</script> denote entities <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><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-525">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-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"><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-571">e_1,e_2</script> denote entities <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>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-572">ent_1 = \rho(e_1), ent_2=\rho(e_2)  \in Entities</script> .
 </li>
-<li><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>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>w</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-526">derivationPath(deriv)= ent_2 \cdot  w \cdot ent_1</script> for some <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>w</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-527">w</script>.
+<li><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>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>w</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-573">derivationPath(deriv)= ent_2 \cdot  w \cdot ent_1</script> for some <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>w</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-574">w</script>.
 </li>
-<li>The attribute values match: <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>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-528">match(W,deriv,attrs)</script>.
+<li>The attribute values match: <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>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-575">match(W,deriv,attrs)</script>.
 </li></ol></div>
 </section>
 </section>
@@ -2313,17 +2378,17 @@
   <h4><span class="secno">4.4.11 </span>Influence</h4>
 
   <div class="semantics" id="influence-semantics" data-count="32" data-title="Semantics 32 (influence-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#influence-semantics">Semantics 32 (influence-semantics)</a></div>
-  <p><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>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>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>o</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>o</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-529">W,\rho \models wasInfluencedBy(id,o_2,o_1,attrs)</script> holds if
+  <p><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>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>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>o</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>o</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-576">W,\rho \models wasInfluencedBy(id,o_2,o_1,attrs)</script> holds if
   and only if at least one of the following hold:</p>
     <ol>
-      <li>[WF] <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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-530">id</script> denotes an influence <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>i</mi><mi>n</mi><mi>f</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>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-531">inf = \rho(id)  \in Influences</script>.
-      </li><li>[WF] <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"><msub><mi>o</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-532">o_1</script> and <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"><msub><mi>o</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-533">o_2</script> denote objects <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>o</mi><mi>b</mi><msub><mi>j</mi><mn>1</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>o</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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-534">obj_1 = \rho(o_1) \in
-  Objects</script> and <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>o</mi><mi>b</mi><msub><mi>j</mi><mn>2</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>o</mi><mn>2</mn></msub><mo stretchy="false">)</mo><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-535">obj_2= \rho(o_2)
+      <li>[WF] <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>i</mi><mi>d</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-577">id</script> denotes an influence <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>i</mi><mi>n</mi><mi>f</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>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-578">inf = \rho(id)  \in Influences</script>.
+      </li><li>[WF] <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"><msub><mi>o</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-579">o_1</script> and <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"><msub><mi>o</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-580">o_2</script> denote objects <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>o</mi><mi>b</mi><msub><mi>j</mi><mn>1</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>o</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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-581">obj_1 = \rho(o_1) \in
+  Objects</script> and <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>o</mi><mi>b</mi><msub><mi>j</mi><mn>2</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>o</mi><mn>2</mn></msub><mo stretchy="false">)</mo><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-582">obj_2= \rho(o_2)
   \in Objects</script>.</li>
-  <li>The influence <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>i</mi><mi>n</mi><mi>f</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-536">inf</script> links <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"><msub><mi>o</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-537">o_2</script> with <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"><msub><mi>o</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-538">o_1</script>; that is,
-  <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>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>n</mi><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>o</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>o</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-539">influenced(inf) = (o_2,o_1)</script>.
+  <li>The influence <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-583-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-583">inf</script> links <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-584-Frame" style="font-size: 100%; "><span class="MathJax_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>o</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-584">o_2</script> with <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-585-Frame" style="font-size: 100%; "><span class="MathJax_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>o</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-585">o_1</script>; that is,
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-586-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>n</mi><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo stretchy="false">(</mo><msub><mi>o</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>o</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-586">influenced(inf) = (o_2,o_1)</script>.
   </li>
-  <li>The attribute values 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>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-540">match(W,deriv,attrs)</script>.
+  <li>The attribute values match: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-587-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-587">match(W,deriv,attrs)</script>.
 </li>
       </ol>
 
@@ -2333,26 +2398,26 @@
 <section id="specialization">
 <h4><span class="secno">4.4.12 </span> Specialization </h4>
 
-<p>The <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>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-541">specializationOf(e_1,e_2)</script> relation indicates when one entity formula presents more specific aspects of another.  
+<p>The <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-588-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-588">specializationOf(e_1,e_2)</script> relation indicates when one entity formula presents more specific aspects of another.  
 </p>
 
     
 <div class="semantics" id="specialization-semantics" data-count="33" data-title="Semantics 33 (specialization-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#specialization-semantics">Semantics 33 (specialization-semantics)</a></div>  <p>
-  <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>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-542">W,\rho \models specializationOf(e_1,e_2)</script> holds if and only if:</p>
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-589-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-589">W,\rho \models specializationOf(e_1,e_2)</script> holds if and only if:</p>
 <ol>
-<li>[WF] Both <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"><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-543">e_1</script> and <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"><msub><mi>e</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-544">e_2</script> are entity identifiers, denoting
-  distinct entities <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>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><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-545">ent_1
-  = \rho(e_1) \in Entities</script> and <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>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-546">ent_2 = \rho(e_2) \in Entities</script>.
+<li>[WF] Both <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-590-Frame" style="font-size: 100%; "><span class="MathJax_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-590">e_1</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-591-Frame" style="font-size: 100%; "><span class="MathJax_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-591">e_2</script> are entity identifiers, denoting
+  entities <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-592-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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-592">ent_1
+  = \rho(e_1) \in Entities</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-593-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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-593">ent_2 = \rho(e_2) \in Entities</script>.
 </li>
 <!--<li>$(ent_1,ent_2) \in SpecializationOf$.-->
-  <li>The two entities present aspects of the same thing, that is, <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>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</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><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-547">thingOf(ent_1) = thingOf(ent_2)</script>.
-</li>
-<li>The lifetime of <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>e</mi><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-548">ent_1</script> is contained in that of <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>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-549">ent_2</script>, i.e. <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>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">⊆</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-550">lifetime(ent_1) \subseteq lifetime(ent_2)</script>.
+  <li>The two entities present aspects of the same thing, that is, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-594-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</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><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-594">thingOf(ent_1) = thingOf(ent_2)</script>.
 </li>
-<li>For each attribute <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>t</mi><mi>t</mi><mi>r</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-551">attr</script> we have <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>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">⊇</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-552">value(ent_1,attr) \supseteq value(ent_2,attr)</script>.</li>
+<li>The lifetime of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-595-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-595">ent_1</script> is contained in that of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-596-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-596">ent_2</script>, i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-597-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">⊆</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-597">lifetime(ent_1) \subseteq lifetime(ent_2)</script>.
+</li>
+<li>For each attribute <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-598-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-598">attr</script> we have <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-599-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">⊇</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-599">value(ent_1,attr) \supseteq value(ent_2,attr)</script>.</li>
 <li>At least one of these inclusions is strict: that is, either
-  <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>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">⊊</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-553">lifetime(ent_1) \subsetneq lifetime(ent_2)</script> or for some <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>t</mi><mi>t</mi><mi>r</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-554">attr</script> we
- value(ent_1,attr) \supsetneq value(ent_2,attr)$.
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-600-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">⊊</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-600">lifetime(ent_1) \subsetneq lifetime(ent_2)</script> or for some <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-601-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-601">attr</script> we have
+ <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-602-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">⊋</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-602">value(ent_1,attr) \supsetneq value(ent_2,attr)</script>.
 </li>
 </ol>
 </div>
@@ -2369,12 +2434,13 @@
 </li></ol>
 -->
 <p>The second criterion says that the two Entities present (possibly different) aspects of
-the same Thing. Note that the third criterion allows <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>e</mi><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-555">ent_1</script> and
-<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>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-556">ent_2</script> to have the same lifetime (or that of <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>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-557">ent_2</script> can be larger).
-The last criterion allows <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>e</mi><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-558">ent_1</script> to have more defined attributes than
-<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>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-559">ent_2</script>, but they must include the attributes defined by <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>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-560">ent_2</script>.  Two
+the same Thing. Note that the third criterion allows <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-603-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-603">ent_1</script> and
+<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-604-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-604">ent_2</script> to have the same lifetime (or that of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-605-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-605">ent_2</script> can be larger).
+The last criterion allows <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-606-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-606">ent_1</script> to have more defined attributes than
+<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-607-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-607">ent_2</script>, but they must include the attributes defined by <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-608-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-608">ent_2</script>.  Two
   different entities that have the same attributes can also be related
-  by specialization.  
+  by specialization.  The fifth criterion (indirectly) ensures that
+  specialization is irreflexive.
 </p>
 </div>
 </section>
@@ -2382,15 +2448,15 @@
 
 <h4><span class="secno">4.4.13 </span> Alternate </h4>
 
-<p>The <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>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-561">alternateOf</script> relation indicates when two entity formulas present (possibly different) aspects of the same thing.  The two entities may or may not overlap in time.
+<p>The <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-609-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-609">alternateOf</script> relation indicates when two entity formulas present (possibly different) aspects of the same thing.  The two entities may or may not overlap in time.
 </p>
 <div class="semantics" id="alternate-semantics" data-count="34" data-title="Semantics 34 (alternate-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#alternate-semantics">Semantics 34 (alternate-semantics)</a></div>
   <p>
-  <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>W</mi><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-562">W,\rho \models alternateOf(e_1,e_2)</script> holds if and only if:</p>
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-610-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-610">W,\rho \models alternateOf(e_1,e_2)</script> holds if and only if:</p>
 <ol>
-<li>[WF] Both <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"><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-563">e_1</script> and <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"><msub><mi>e</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-564">e_2</script> are entity identifiers, denoting <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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-565">ent_1 = \rho(e_1)</script> and <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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-566">ent_2 = \rho(e_2)</script>.
+<li>[WF] Both <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-611-Frame" style="font-size: 100%; "><span class="MathJax_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-611">e_1</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-612-Frame" style="font-size: 100%; "><span class="MathJax_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-612">e_2</script> are entity identifiers, denoting <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-613-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-613">ent_1 = \rho(e_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-614-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-614">ent_2 = \rho(e_2)</script>.
 </li>
-<li>The two objects refer to the same underlying Thing: <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>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</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><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-567">thingOf(ent_1) = thingOf(ent_2)</script>
+<li>The two objects refer to the same underlying Thing: <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-615-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</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><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-615">thingOf(ent_1) = thingOf(ent_2)</script>
 </li></ol>
 </div>
 
@@ -2402,17 +2468,17 @@
 
 <h4><span class="secno">4.4.14 </span> Membership </h4>
 
-<p>The <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>h</mi><mi>a</mi><mi>d</mi><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-568">hadMember</script> relation relates a collection to an element of the collection.
+<p>The <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-616-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mi>a</mi><mi>d</mi><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-616">hadMember</script> relation relates a collection to an element of the collection.
 </p>
 <div class="semantics" id="membership-semantics" data-count="35" data-title="Semantics 35 (membership-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#membership-semantics">Semantics 35 (membership-semantics)</a></div>
   <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>h</mi><mi>a</mi><mi>d</mi><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-569">W,\rho \models hadMember(c,e)</script> holds if and only if:</p>
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-617-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>h</mi><mi>a</mi><mi>d</mi><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-617">W,\rho \models hadMember(c,e)</script> holds if and only if:</p>
 <ol>
-<li>[WF] Both <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"><msub><mi>e</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-570">e_1</script> and <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"><msub><mi>e</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-571">e_2</script> are entity identifiers, denoting <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>c</mi><mi>o</mi><mi>l</mi><mi>l</mi><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-572">coll
-  = \rho(c) \in Collections</script> and <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><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-573">ent = \rho(e) \in Entities</script>.
+<li>[WF] Both <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-618-Frame" style="font-size: 100%; "><span class="MathJax_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-618">e_1</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-619-Frame" style="font-size: 100%; "><span class="MathJax_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-619">e_2</script> are entity identifiers, denoting <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-620-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>l</mi><mi>l</mi><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-620">coll
+  = \rho(c) \in Collections</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-621-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-621">ent = \rho(e) \in Entities</script>.
 </li>
-<li>The entity <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>e</mi><mi>n</mi><mi>t</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-574">ent</script> is a member of the collection <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>c</mi><mi>o</mi><mi>l</mi><mi>l</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-575">coll</script>: that is,
-<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"><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">,</mo><mi>c</mi><mi>o</mi><mi>l</mi><mi>l</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>O</mi><mi>f</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-576">(ent,coll) \in MemberOf</script>.
+<li>The entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-622-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-622">ent</script> is a member of the collection <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-623-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>l</mi><mi>l</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-623">coll</script>: that is,
+<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-624-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">,</mo><mi>c</mi><mi>o</mi><mi>l</mi><mi>l</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>O</mi><mi>f</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-624">(ent,coll) \in MemberOf</script>.
 </li></ol>
 </div>
 
@@ -2431,25 +2497,25 @@
   
   <section id="precedes-and-strictly-precedes">
   <h4><span class="secno">4.5.1 </span>Precedes and Strictly Precedes</h4>
-  <p>The precedes relation <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>x</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-577">x \precedes y</script> holds between two events, one taking
+  <p>The precedes relation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-625-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-625">x \precedes y</script> holds between two events, one taking
   place before (or simultaneously with) another.  Its meaning is
   defined in terms of the quasiordering on events specified by
-  <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"><mo stretchy="false">⪯</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-578">\preceq</script>.  The semantics of strictly precedes (<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>x</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">strictlyPrecedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-579">x
-  \strictlyPrecedes y</script>) is similar, only <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>x</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-580">x</script> must take place strictly
-  before <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>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-581">y</script>.  It is interpreted as <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"><mo stretchy="false">≺</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-582">\prec</script>, which we recall is
-  defined from <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-583-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">⪯</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-583">\preceq</script> as <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-584-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">≺</mo><mi>y</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">⟺</mo><mspace width="thickmathspace"></mspace><mi>x</mi><mo stretchy="false">⪯</mo><mi>y</mi><mtext>&nbsp;and&nbsp;</mtext><mi>y</mi><mo stretchy="false">⪯̸</mo><mi>x</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-584">x \prec y \iff x \preceq y \text{ and } y
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-626-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">⪯</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-626">\preceq</script>.  The semantics of strictly precedes (<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-627-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">strictlyPrecedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-627">x
+  \strictlyPrecedes y</script>) is similar, only <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-628-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-628">x</script> must take place strictly
+  before <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-629-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-629">y</script>.  It is interpreted as <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-630-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">≺</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-630">\prec</script>, which we recall is
+  defined from <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-631-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">⪯</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-631">\preceq</script> as <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-632-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">≺</mo><mi>y</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">⟺</mo><mspace width="thickmathspace"></mspace><mi>x</mi><mo stretchy="false">⪯</mo><mi>y</mi><mtext>&nbsp;and&nbsp;</mtext><mi>y</mi><mo stretchy="false">⪯̸</mo><mi>x</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-632">x \prec y \iff x \preceq y \text{ and } y
   \not\preceq x</script>.</p>
   <div class="semantics" id="precedes-semantics" data-count="36" data-title="Semantics 36 (precedes-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#precedes-semantics">Semantics 36 (precedes-semantics)</a></div>
     <ol>
-      <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-585-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-585">W,\rho \models x \precedes y</script> holds if and only if
-      <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-586-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">∈</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-586">\rho(x),\rho(y) \in Events</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-587-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-587">\rho(x) \preceq\rho(y)</script>.</li>
-      <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-588-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">strictlyPrecedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-588">W,\rho \models x \strictlyPrecedes y</script> holds if and only if
-       <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-589-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">∈</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-589">\rho(x),\rho(y) \in Events</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-590-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-590">\rho(x) \prec \rho(y)</script>.</li>
+      <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-633-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-633">W,\rho \models x \precedes y</script> holds if and only if
+      <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-634-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">∈</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-634">\rho(x),\rho(y) \in Events</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-635-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-635">\rho(x) \preceq\rho(y)</script>.</li>
+      <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-636-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">strictlyPrecedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>y</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-636">W,\rho \models x \strictlyPrecedes y</script> holds if and only if
+       <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-637-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">∈</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-637">\rho(x),\rho(y) \in Events</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-638-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-638">\rho(x) \prec \rho(y)</script>.</li>
     </ol>
   </div>
 
   <div class="remark">
-    <p> Although times are linearly ordered, the time ordering is
+    <p> The ordering of time values associated to events is
     unrelated to the event ordering. For example:</p>
     <pre>entity(e)
 activity(a1)
@@ -2458,7 +2524,7 @@
 wasGeneratedBy(gen2; e, a2, 2012-11-16T16:05:00) //different date
 </pre>
     <p>This instance is valid, and must satisfy precedence constraints
-    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-591-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><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-591">gen_1 \precedes gen_2</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-592-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>2</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-592">gen_2 \precedes gen_1</script>, but this does not
+    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-639-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><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-639">gen_1 \precedes gen_2</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-640-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>2</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-640">gen_2 \precedes gen_1</script>, but this does not
     imply anything about the relative orderings of the associated
     times, or vice versa.
     </p>
@@ -2468,11 +2534,11 @@
   
   <section id="notnull">
   <h4><span class="secno">4.5.2 </span>notNull</h4>
-  <p>The <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-593-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-593">notNull(x)</script> formula is used to specify that a value may not
-  be the null value <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-594-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">⊥</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-594">\bot</script>.  The symbol <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-595-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">−</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-595">-</script> always denotes the null
-  value (i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-596-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">−</mo><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="normal">⊥</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-596">\rho(-) = \bot</script>).
+  <p>The <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-641-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-641">notNull(x)</script> formula is used to specify that a value may not
+  be the null value <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-642-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">⊥</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-642">\bot</script>.  The symbol <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-643-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">−</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-643">-</script> always denotes the null
+  value (i.e. <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-644-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">−</mo><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi mathvariant="normal">⊥</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-644">\rho(-) = \bot</script>).
   </p><div class="semantics" id="notNull-semantics" data-count="37" data-title="Semantics 37 (notNull-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#notNull-semantics">Semantics 37 (notNull-semantics)</a></div>
-    <p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-597-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-597">W,\rho\models notNull(e)</script> holds if and only if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-598-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">)</mo><mo stretchy="false">≠</mo><mi mathvariant="normal">⊥</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-598">\rho(e) \neq
+    <p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-645-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-645">W,\rho\models notNull(e)</script> holds if and only if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-646-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">)</mo><mo stretchy="false">≠</mo><mi mathvariant="normal">⊥</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-646">\rho(e) \neq
   \bot</script>.
       </p>
   </div> 
@@ -2482,21 +2548,21 @@
   <section id="typeof">
   <h4><span class="secno">4.5.3 </span>typeOf</h4>
   
-  <p>The typing formula <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-599-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-599">typeOf(x,t)</script> constrains the type of the value of
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-600-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-600">x</script>.  </p>
+  <p>The typing formula <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-647-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-647">typeOf(x,t)</script> constrains the type of the value of
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-648-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-648">x</script>.  </p>
 
   <div class="semantics" id="typeOf-semantics" data-count="38" data-title="Semantics 38 (typeOf-semantics)"><div class="ruleTitle"><a class="internalDFN" href="#typeOf-semantics">Semantics 38 (typeOf-semantics)</a></div>
-    <ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-601-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-601">W,\rho\models typeOf(e,entity)</script> holds if and only if
-      <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-602-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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-602">\rho(e) \in Entities</script>.</li>
-      <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-603-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><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>y</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-603">W,\rho\models typeOf(a,activity)</script> holds if and only if
-      <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-604-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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-604">\rho(a) \in Activities</script>.</li>
-      <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-605-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mi>g</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-605">W,\rho\models typeOf(ag,agent)</script> holds if and only if
-      <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-606-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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-606">\rho(ag) \in Agents</script>.</li>
-      <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-607-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-607">W,\rho\models typeOf(c,Collection)</script> holds if and only if
-      <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-608-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>c</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-608">\rho(c) \in Collections</script>.</li>
-      <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-609-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>E</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-609">W,\rho\models typeOf(c,EmptyCollection)</script> holds if and only if
-      <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-610-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>c</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-610">\rho(c) \in Collections</script> and there is no <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-611-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-611">e \in Entities</script> such
-    that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-612-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>O</mi><mi>f</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-612">(e,\rho(c)) \in MemberOf</script>.</li>
+    <ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-649-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</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></math></span></span></span><script type="math/tex" id="MathJax-Element-649">W,\rho\models typeOf(e,entity)</script> holds if and only if
+      <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-650-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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-650">\rho(e) \in Entities</script>.</li>
+      <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-651-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><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>y</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-651">W,\rho\models typeOf(a,activity)</script> holds if and only if
+      <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-652-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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-652">\rho(a) \in Activities</script>.</li>
+      <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-653-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mi>g</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-653">W,\rho\models typeOf(ag,agent)</script> holds if and only if
+      <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-654-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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-654">\rho(ag) \in Agents</script>.</li>
+      <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-655-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-655">W,\rho\models typeOf(c,Collection)</script> holds if and only if
+      <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-656-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>c</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-656">\rho(c) \in Collections</script>.</li>
+      <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-657-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>E</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-657">W,\rho\models typeOf(c,EmptyCollection)</script> holds if and only if
+      <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-658-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>c</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-658">\rho(c) \in Collections</script> and there is no <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-659-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-659">e \in Entities</script> such
+    that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-660-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>O</mi><mi>f</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-660">(e,\rho(c)) \in MemberOf</script>.</li>
       </ol>
       
     </div>
@@ -2520,7 +2586,7 @@
 <section id="inferences">
 <h3><span class="secno">5.1 </span>Inferences</h3>
 
-<div class="inference" id="communication-generation-use-inference"><div class="ruleTitle"><a class="internalDFN" href="#communication-generation-use-inference">Inference 5 (communication-generation-use-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-613-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi mathvariant="normal">∃</mi><mi>e</mi><mo stretchy="false">,</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>u</mi><mi>s</mi><mi>e</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext><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>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</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><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><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-613">\begin{array}[t]{l}
+<div class="inference" id="communication-generation-use-inference"><div class="ruleTitle"><a class="internalDFN" href="#communication-generation-use-inference">Inference 5 (communication-generation-use-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-661-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi mathvariant="normal">∃</mi><mi>e</mi><mo stretchy="false">,</mo><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>u</mi><mi>s</mi><mi>e</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext><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>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</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><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><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-661">\begin{array}[t]{l}
 \forall id,a_2,a_1,attrs.~
 \\
 \qquad wasInformedBy(id,a_2,a_1,attrs)
@@ -2530,9 +2596,9 @@
 \end{array}</script></div>
 
 <div class="proof">
-<p> This follows immediately from the semantics of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-614-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-614">wasInformedBy</script>.</p>
+<p> This follows immediately from the semantics of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-662-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-662">wasInformedBy</script>.</p>
   </div>
-<div class="inference" id="generation-use-communication-inference"><div class="ruleTitle"><a class="internalDFN" href="#generation-use-communication-inference">Inference 6 (generation-use-communication-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-615-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>u</mi><mi>s</mi><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><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><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi mathvariant="normal">∃</mi><mi>i</mi><mi>d</mi><mo>.</mo><mtext>&nbsp;</mtext><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><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-615">\begin{array}[t]{l}
+<div class="inference" id="generation-use-communication-inference"><div class="ruleTitle"><a class="internalDFN" href="#generation-use-communication-inference">Inference 6 (generation-use-communication-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-663-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>u</mi><mi>s</mi><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><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><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi mathvariant="normal">∃</mi><mi>i</mi><mi>d</mi><mo>.</mo><mtext>&nbsp;</mtext><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><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-663">\begin{array}[t]{l}
 \forall gen,e,a_1,t_1,attrs_1,use,a_2,t_2,attrs_2.~
 \\
 \qquad wasGeneratedBy(gen,e,a_1,t_1,attrs_1) \wedge used(use,a_2,e,t_2,attrs_2)
@@ -2541,9 +2607,9 @@
 \exists id.~wasInformedBy(id,a_2,a_1,[])
 \end{array}</script></div>
 <div class="proof">
-<p> This follows from the semantics of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-616-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-616">wasInformedBy</script> and <a href="#axiom1">Axiom 1</a>.</p>
+<p> This follows from the semantics of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-664-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-664">wasInformedBy</script> and <a href="#axiom1">Axiom 1</a>.</p>
   </div>
-<div class="inference" id="entity-generation-invalidation-inference"><div class="ruleTitle"><a class="internalDFN" href="#entity-generation-invalidation-inference">Inference 7 (entity-generation-invalidation-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-617-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi mathvariant="normal">∃</mi><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>i</mi><mi>n</mi><mi>v</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext><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>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</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><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>n</mi><mi>v</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-617">\begin{array}[t]{l}
+<div class="inference" id="entity-generation-invalidation-inference"><div class="ruleTitle"><a class="internalDFN" href="#entity-generation-invalidation-inference">Inference 7 (entity-generation-invalidation-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-665-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi mathvariant="normal">∃</mi><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>i</mi><mi>n</mi><mi>v</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext><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>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</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><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>n</mi><mi>v</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-665">\begin{array}[t]{l}
 \forall e,attrs.~
 \\
 \qquad entity(e,attrs)
@@ -2557,7 +2623,7 @@
   the requirement that generation and invalidation events exist for
   the entity.</p>
   </div>
-<div class="inference" id="activity-start-end-inference"><div class="ruleTitle"><a class="internalDFN" href="#activity-start-end-inference">Inference 8 (activity-start-end-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-618-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi mathvariant="normal">∃</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</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><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><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><msub><mi>t</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><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>e</mi><mi>n</mi><mi>d</mi><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-618">\begin{array}[t]{l}
+<div class="inference" id="activity-start-end-inference"><div class="ruleTitle"><a class="internalDFN" href="#activity-start-end-inference">Inference 8 (activity-start-end-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-666-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi mathvariant="normal">∃</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</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><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><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><msub><mi>t</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><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>e</mi><mi>n</mi><mi>d</mi><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-666">\begin{array}[t]{l}
 \forall a,t_1,t_2,attrs.~
 \\
 \qquad activity(a,t_1,t_2,attrs)
@@ -2573,7 +2639,7 @@
   </div>
 
   
-<div class="inference" id="wasStartedBy-inference"><div class="ruleTitle"><a class="internalDFN" href="#wasStartedBy-inference">Inference 9 (wasStartedBy-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-619-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><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><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><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><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi mathvariant="normal">∃</mi><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo>.</mo><mtext>&nbsp;</mtext><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>g</mi><mi>e</mi><mi>n</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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-619">\begin{array}[t]{l}
+<div class="inference" id="wasStartedBy-inference"><div class="ruleTitle"><a class="internalDFN" href="#wasStartedBy-inference">Inference 9 (wasStartedBy-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-667-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><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><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><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><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi mathvariant="normal">∃</mi><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo>.</mo><mtext>&nbsp;</mtext><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>g</mi><mi>e</mi><mi>n</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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-667">\begin{array}[t]{l}
 \forall id,a,e_1,a_1,t,attrs.~
 \\
 \qquad wasStartedBy(id,a,e_1,a_1,t,attrs)
@@ -2585,7 +2651,7 @@
 <div class="proof">
   <p>This follows from <a href="#axiom2">Axiom 2</a>.</p>
   </div>
-<div class="inference" id="wasEndedBy-inference"><div class="ruleTitle"><a class="internalDFN" href="#wasEndedBy-inference">Inference 10 (wasEndedBy-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-620-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><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><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><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><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi mathvariant="normal">∃</mi><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo>.</mo><mtext>&nbsp;</mtext><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>g</mi><mi>e</mi><mi>n</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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-620">\begin{array}[t]{l}
+<div class="inference" id="wasEndedBy-inference"><div class="ruleTitle"><a class="internalDFN" href="#wasEndedBy-inference">Inference 10 (wasEndedBy-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-668-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><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><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><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><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi mathvariant="normal">∃</mi><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo>.</mo><mtext>&nbsp;</mtext><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>g</mi><mi>e</mi><mi>n</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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-668">\begin{array}[t]{l}
 \forall id,a,e_1,a_1,t,attrs.~
 \\
 \qquad wasEndedBy(id,a,e_1,a_1,t,attrs)
@@ -2598,7 +2664,7 @@
   <p>This follows from <a href="#axiom3">Axiom 3</a>.</p>
   </div>
 <div class="inference" id="derivation-generation-use-inference"><div class="ruleTitle"><a class="internalDFN" href="#derivation-generation-use-inference">Inference 11 (derivation-generation-use-inference)</a></div>
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-621-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>u</mi><mi>s</mi><msub><mi>e</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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>u</mi><mi>s</mi><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><mo stretchy="false">,</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>u</mi><mi>s</mi><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi mathvariant="normal">∃</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>u</mi><mi>s</mi><msub><mi>e</mi><mn>1</mn></msub><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>t</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><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-621">\begin{array}[t]{l}
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-669-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>u</mi><mi>s</mi><msub><mi>e</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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>u</mi><mi>s</mi><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><mo stretchy="false">,</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>u</mi><mi>s</mi><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi mathvariant="normal">∃</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>u</mi><mi>s</mi><msub><mi>e</mi><mn>1</mn></msub><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>t</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><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-669">\begin{array}[t]{l}
 \forall id,e_2,e_1,a,gen_2,use_1,attrs.~
 \\
 \qquad notNull(a) \wedge notNull(gen_2) \wedge notNull(use_1) \wedge wasDerivedFrom(id,e_2,e_1,a,gen_2,use_1,attrs)
@@ -2612,7 +2678,7 @@
   </div>
   
 <div class="inference" id="revision-is-alternate-inference"><div class="ruleTitle"><a class="internalDFN" href="#revision-is-alternate-inference">Inference 12 (revision-is-alternate-inference)</a></div>
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-622-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>g</mi><mo stretchy="false">,</mo><mi>u</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><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><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><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-622">\begin{array}[t]{l}
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-670-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>g</mi><mo stretchy="false">,</mo><mi>u</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><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><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><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-670">\begin{array}[t]{l}
 \forall id,e_1,e_2,a,g,u.~
 \\
 \qquad wasDerivedFrom(id,e_2,e_1,a,g,u,[prov:type = prov:Revision]))
@@ -2624,7 +2690,7 @@
   <p> This follows from the semantics of derivation steps (precise or
   imprecise) and <a href="#axiom4">Axiom 4</a>.</p>
   </div>
-<div class="inference" id="attribution-inference"><div class="ruleTitle"><a class="internalDFN" href="#attribution-inference">Inference 13 (attribution-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-623-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>a</mi><mi>t</mi><mi>t</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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>a</mi><mi>t</mi><mi>t</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi mathvariant="normal">∃</mi><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>g</mi><mi>e</mi><mi>n</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>p</mi><mi>l</mi><mo>.</mo><mtext>&nbsp;</mtext><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>g</mi><mi>e</mi><mi>n</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><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><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>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</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><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-623">\begin{array}[t]{l}
+<div class="inference" id="attribution-inference"><div class="ruleTitle"><a class="internalDFN" href="#attribution-inference">Inference 13 (attribution-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-671-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>a</mi><mi>t</mi><mi>t</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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>a</mi><mi>t</mi><mi>t</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi mathvariant="normal">∃</mi><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>g</mi><mi>e</mi><mi>n</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>p</mi><mi>l</mi><mo>.</mo><mtext>&nbsp;</mtext><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>g</mi><mi>e</mi><mi>n</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><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><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>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</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><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-671">\begin{array}[t]{l}
 \forall att,e,ag,attrs.~
 \\
 \qquad wasAttributedTo(att,e,ag,attrs)
@@ -2634,10 +2700,10 @@
 \end{array}</script></div>
 <div class="proof">
   <p>This follows from the semantics of generation, association, and
-  attribution, by <a href="#axiom5">Axiom 5</a></p>
+  attribution, by <a href="#axiom5">Axiom 5</a>.</p>
   </div>
   
-<div class="inference" id="delegation-inference"><div class="ruleTitle"><a class="internalDFN" href="#delegation-inference">Inference 14 (delegation-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-624-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><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><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi mathvariant="normal">∃</mi><mi>i</mi><msub><mi>d</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>i</mi><msub><mi>d</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext><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><msub><mi>d</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><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><msub><mi>d</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-624">\begin{array}[t]{l}
+<div class="inference" id="delegation-inference"><div class="ruleTitle"><a class="internalDFN" href="#delegation-inference">Inference 14 (delegation-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-672-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><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><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi mathvariant="normal">∃</mi><mi>i</mi><msub><mi>d</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>i</mi><msub><mi>d</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext><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><msub><mi>d</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo><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><msub><mi>d</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>p</mi><msub><mi>l</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">[</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-672">\begin{array}[t]{l}
 \forall id,ag_1,ag_2,a,attrs.~
 \\
 \qquad actedOnBehalfOf(id,ag_1,ag_2,a,attrs)
@@ -2646,51 +2712,51 @@
 \exists id_1,pl_1,id_2,pl_2.~wasAssociatedWith(id_1,a,ag_1,pl_1,[]) \wedge wasAssociatedWith(id_2,a,ag_2,pl_2,[])
 \end{array}</script></div>
 <div class="proof">
-  <p>This follows from the semantics of association and delegation, by <a href="#axiom6">Axiom 6</a></p>
+  <p>This follows from the semantics of association and delegation, by <a href="#axiom6">Axiom 6</a>.</p>
   </div>
-<div class="inference" id="influence-inference"><div class="ruleTitle"><a class="internalDFN" href="#influence-inference">Inference 15 (influence-inference)</a></div><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-625-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>w</mi><mi>a</mi><mi>s</mi><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-625">\begin{array}[t]{l}
+<div class="inference" id="influence-inference"><div class="ruleTitle"><a class="internalDFN" href="#influence-inference">Inference 15 (influence-inference)</a></div><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-673-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>w</mi><mi>a</mi><mi>s</mi><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-673">\begin{array}[t]{l}
 \forall id,e,a,t,attrs.~
 \\
 \qquad wasGeneratedBy(id,e,a,t,attrs)
 \\
 \quad\Rightarrow
 wasInfluencedBy(id,e,a,attrs)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-626-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-626">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-674-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-674">\begin{array}[t]{l}
 \forall id,a,e,t,attrs.~
 \\
 \qquad used(id,a,e,t,attrs)
 \\
 \quad\Rightarrow
 wasInfluencedBy(id,a,e,attrs)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-627-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-627">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-675-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-675">\begin{array}[t]{l}
 \forall id,a_2,a_1,attrs.~
 \\
 \qquad wasInformedBy(id,a_2,a_1,attrs)
 \\
 \quad\Rightarrow
 wasInfluencedBy(id,a_2,a_1,attrs)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-628-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>w</mi><mi>a</mi><mi>s</mi><mi>S</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-628">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-676-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>w</mi><mi>a</mi><mi>s</mi><mi>S</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-676">\begin{array}[t]{l}
 \forall id,a_2,e,a_1,t,attrs.~
 \\
 \qquad wasStartedBy(id,a_2,e,a_1,t,attrs)
 \\
 \quad\Rightarrow
 wasInfluencedBy(id,a_2,e,attrs)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-629-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>w</mi><mi>a</mi><mi>s</mi><mi>E</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-629">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-677-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>w</mi><mi>a</mi><mi>s</mi><mi>E</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-677">\begin{array}[t]{l}
 \forall id,a_2,e,a_1,t,attrs.~
 \\
 \qquad wasEndedBy(id,a_2,e,a_1,t,attrs)
 \\
 \quad\Rightarrow
 wasInfluencedBy(id,a_2,e,attrs)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-630-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-630">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-678-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-678">\begin{array}[t]{l}
 \forall id,e,a,t,attrs.~
 \\
 \qquad wasInvalidatedBy(id,e,a,t,attrs)
 \\
 \quad\Rightarrow
 wasInfluencedBy(id,e,a,attrs)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-631-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><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>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-631">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-679-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><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>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-679">\begin{array}[t]{l}
 \forall id,e_2,e_1,a,g,u,attrs.~
 \\
 \qquad wasDerivedFrom(id,e_2,e_1,a,g,u,attrs)
@@ -2698,7 +2764,7 @@
 \quad\Rightarrow
 wasInfluencedBy(id,e_2,e_1,attrs)
 \end{array}</script></li><li>
-<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-632-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>T</mi><mi>o</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-632">\begin{array}[t]{l}
+<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-680-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>T</mi><mi>o</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-680">\begin{array}[t]{l}
 \forall id,e,ag,attrs.~
 \\
 \qquad wasAttributedTo(id,e,ag,attrs)
@@ -2706,14 +2772,14 @@
 \quad\Rightarrow
 wasInfluencedBy(id,e,ag,attrs)
 \end{array}</script></li><li>
-<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-633-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-633">\begin{array}[t]{l}
+<span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-681-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-681">\begin{array}[t]{l}
 \forall id,a,ag,pl,attrs.~
 \\
 \qquad wasAssociatedWith(id,a,ag,pl,attrs)
 \\
 \quad\Rightarrow
 wasInfluencedBy(id,a,ag,attrs)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-634-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>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>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-634">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-682-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>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>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-682">\begin{array}[t]{l}
 \forall id,ag_2,ag_1,a,attrs.~
 \\
 \qquad actedOnBehalfOf(id,ag_2,ag_1,a,attrs)
@@ -2725,7 +2791,7 @@
 <div class="proof">
   <p>This follows via <a href="#axiom7">Axioms 7</a> through <a href="#axiom16">16</a>.
   </p></div>
-<div class="inference" id="alternate-reflexive"><div class="ruleTitle"><a class="internalDFN" href="#alternate-reflexive">Inference 16 (alternate-reflexive)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-635-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>e</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-635">\begin{array}[t]{l}
+<div class="inference" id="alternate-reflexive"><div class="ruleTitle"><a class="internalDFN" href="#alternate-reflexive">Inference 16 (alternate-reflexive)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-683-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>e</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-683">\begin{array}[t]{l}
 \forall e.~
 \\
 \qquad entity(e)
@@ -2734,10 +2800,10 @@
 alternateOf(e,e)
 \end{array}</script></div>
 <div class="proof">
-  <p> Suppose <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-636-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-636">ent = \rho(e)</script>.  Clearly <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-637-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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-637">ent \in Entities</script> and
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-638-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><mi>t</mi><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><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-638">thingOf(ent) = thingOf(ent)</script>, so <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-639-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-639">W,\rho \models alternateOf(e,e)</script>.</p>
+  <p> Suppose <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-684-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-684">ent = \rho(e)</script>.  Clearly <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-685-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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-685">ent \in Entities</script> and
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-686-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><mi>t</mi><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><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-686">thingOf(ent) = thingOf(ent)</script>, so <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-687-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-687">W,\rho \models alternateOf(e,e)</script>.</p>
   </div>
-<div class="inference" id="alternate-transitive"><div class="ruleTitle"><a class="internalDFN" href="#alternate-transitive">Inference 17 (alternate-transitive)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-640-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>3</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-640">\begin{array}[t]{l}
+<div class="inference" id="alternate-transitive"><div class="ruleTitle"><a class="internalDFN" href="#alternate-transitive">Inference 17 (alternate-transitive)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-688-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>3</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-688">\begin{array}[t]{l}
 \forall e_1,e_2,e_3.~
 \\
 \qquad alternateOf(e_1,e_2) \wedge alternateOf(e_2,e_3)
@@ -2746,13 +2812,13 @@
 alternateOf(e_1,e_3)
 \end{array}</script></div>
 <div class="proof">
-  <p> Suppose <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-641-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-641">ent_1 = \rho(e_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-642-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-642">ent_2 = \rho(e_2)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-643-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>3</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>3</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-643">ent_3 = \rho(e_3)</script>.  Then  by
-  assumption <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-644-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-644">ent_1</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-645-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-645">ent_2</script>, and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-646-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>3</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-646">ent_3</script> are in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-647-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-647">Entities</script> and
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-648-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-648">thingOf(e_1) = thingOf(e_2)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-649-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>2</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>3</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-649">thingOf(e_2) = thingOf(e_3)</script>, so
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-650-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>3</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-650">thingOf(e_1) = thingOf(e_3)</script>, as required to conclude <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-651-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>3</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-651">W,\rho
+  <p> Suppose <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-689-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-689">ent_1 = \rho(e_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-690-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-690">ent_2 = \rho(e_2)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-691-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>3</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>3</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-691">ent_3 = \rho(e_3)</script>.  Then  by
+  assumption <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-692-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-692">ent_1</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-693-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-693">ent_2</script>, and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-694-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>3</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-694">ent_3</script> are in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-695-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-695">Entities</script> and
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-696-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-696">thingOf(e_1) = thingOf(e_2)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-697-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>2</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>3</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-697">thingOf(e_2) = thingOf(e_3)</script>, so
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-698-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>3</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-698">thingOf(e_1) = thingOf(e_3)</script>, as required to conclude <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-699-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>3</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-699">W,\rho
   \models alternateOf(e_1,e_3)</script>.</p>
   </div>
-<div class="inference" id="alternate-symmetric"><div class="ruleTitle"><a class="internalDFN" href="#alternate-symmetric">Inference 18 (alternate-symmetric)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-652-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-652">\begin{array}[t]{l}
+<div class="inference" id="alternate-symmetric"><div class="ruleTitle"><a class="internalDFN" href="#alternate-symmetric">Inference 18 (alternate-symmetric)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-700-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-700">\begin{array}[t]{l}
 \forall e_1,e_2.~
 \\
 \qquad alternateOf(e_1,e_2)
@@ -2761,13 +2827,13 @@
 alternateOf(e_2,e_1)
 \end{array}</script></div>
 <div class="proof">
-  <p> Suppose <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-653-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-653">ent_1 = \rho(e_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-654-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-654">ent_2 = \rho(e_2)</script>.  Then  by
-  assumption both <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-655-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-655">ent_1</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-656-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-656">ent_2</script> are in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-657-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-657">Entities</script> and
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-658-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-658">thingOf(e_1) = thingOf(e_2)</script>, as required to conclude <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-659-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>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-659">W,\rho
+  <p> Suppose <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-701-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-701">ent_1 = \rho(e_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-702-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-702">ent_2 = \rho(e_2)</script>.  Then  by
+  assumption both <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-703-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-703">ent_1</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-704-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-704">ent_2</script> are in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-705-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-705">Entities</script> and
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-706-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-706">thingOf(e_1) = thingOf(e_2)</script>, as required to conclude <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-707-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>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-707">W,\rho
   \models alternateOf(e_2,e_1)</script>.</p>
   </div>
   
-<div class="inference" id="specialization-transitive"><div class="ruleTitle"><a class="internalDFN" href="#specialization-transitive">Inference 19 (specialization-transitive)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-660-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>3</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-660">\begin{array}[t]{l}
+<div class="inference" id="specialization-transitive"><div class="ruleTitle"><a class="internalDFN" href="#specialization-transitive">Inference 19 (specialization-transitive)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-708-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>3</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-708">\begin{array}[t]{l}
 \forall e_1,e_2,e_3.~
 \\
 \qquad specializationOf(e_1,e_2) \wedge specializationOf(e_2,e_3)
@@ -2776,20 +2842,20 @@
 specializationOf(e_1,e_3)
 \end{array}</script></div>
 <div class="proof">
-  <p> Suppose the conditions for specialization hold of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-661-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-661">ent_1</script> and
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-662-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-662">ent_2</script> and for <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-663-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-663">ent_2</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-664-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>3</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-664">ent_3</script>, where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-665-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-665">ent_1 = \rho(e_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-666-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-666">ent_2 = \rho(e_2)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-667-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>3</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>3</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-667">ent_3 =
-  \rho(e_3)</script>. Then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-668-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">⊆</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">⊆</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>3</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-668">lifetime(e_1) \subseteq lifetime(e_2) \subseteq
+  <p> Suppose the conditions for specialization hold of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-709-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-709">ent_1</script> and
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-710-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-710">ent_2</script> and for <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-711-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-711">ent_2</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-712-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>3</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-712">ent_3</script>, where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-713-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-713">ent_1 = \rho(e_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-714-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-714">ent_2 = \rho(e_2)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-715-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>3</mn></msub><mo stretchy="false">=</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>3</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-715">ent_3 =
+  \rho(e_3)</script>. Then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-716-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">⊆</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">⊆</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>3</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-716">lifetime(e_1) \subseteq lifetime(e_2) \subseteq
   lifetime(e_3)</script>.  Moreover, 
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-669-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>o</mi><mi>b</mi><msub><mi>j</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">⊇</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>o</mi><mi>b</mi><msub><mi>j</mi><mn>3</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-669">value(obj_2,attr) \supseteq value(obj_3,attr)</script>, and similarly
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-670-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>o</mi><mi>b</mi><msub><mi>j</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">⊇</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>o</mi><mi>b</mi><msub><mi>j</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-670">value(obj_1,attr)\supseteq value(obj_2,attr)</script> so <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-671-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>o</mi><mi>b</mi><msub><mi>j</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">⊇</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>o</mi><mi>b</mi><msub><mi>j</mi><mn>3</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-671">value(obj_1,attr)
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-717-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>o</mi><mi>b</mi><msub><mi>j</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">⊇</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>o</mi><mi>b</mi><msub><mi>j</mi><mn>3</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-717">value(obj_2,attr) \supseteq value(obj_3,attr)</script>, and similarly
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-718-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>o</mi><mi>b</mi><msub><mi>j</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">⊇</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>o</mi><mi>b</mi><msub><mi>j</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-718">value(obj_1,attr)\supseteq value(obj_2,attr)</script> so <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-719-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>o</mi><mi>b</mi><msub><mi>j</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">⊇</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>o</mi><mi>b</mi><msub><mi>j</mi><mn>3</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-719">value(obj_1,attr)
   \supseteq value(obj_3,attr)</script>.  Finally, at least one of the
-  inclusions between <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-672-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>j</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-672">obj_1</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-673-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>j</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-673">obj_2</script> is strict, so the same is the
-  case for <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-674-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>j</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-674">obj_1</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-675-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>j</mi><mn>3</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-675">obj_3</script>.
+  inclusions between <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-720-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>j</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-720">obj_1</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-721-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>j</mi><mn>2</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-721">obj_2</script> is strict, so the same is the
+  case for <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-722-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>j</mi><mn>1</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-722">obj_1</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-723-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>j</mi><mn>3</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-723">obj_3</script>.
 </p>
 </div>
 
 
-<div class="inference" id="specialization-alternate-inference"><div class="ruleTitle"><a class="internalDFN" href="#specialization-alternate-inference">Inference 20 (specialization-alternate-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-676-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-676">\begin{array}[t]{l}
+<div class="inference" id="specialization-alternate-inference"><div class="ruleTitle"><a class="internalDFN" href="#specialization-alternate-inference">Inference 20 (specialization-alternate-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-724-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-724">\begin{array}[t]{l}
 \forall e_1,e_2.~
 \\
 \qquad specializationOf(e_1,e_2)
@@ -2798,12 +2864,12 @@
 alternateOf(e_1,e_2)
 \end{array}</script></div>
 <div class="proof">
-  <p> If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-677-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-677">ent_1=\rho(e_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-678-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-678">ent_2 = \rho(e_2)</script> are
-  specializations, then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-679-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</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><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-679">thingOf(ent_1) = thingOf(ent_2)</script>.
+  <p> If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-725-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-725">ent_1=\rho(e_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-726-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-726">ent_2 = \rho(e_2)</script> are
+  specializations, then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-727-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</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><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-727">thingOf(ent_1) = thingOf(ent_2)</script>.
 </p>
 </div>
 
-<div class="inference" id="specialization-attributes-inference"><div class="ruleTitle"><a class="internalDFN" href="#specialization-attributes-inference">Inference 21 (specialization-attributes-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-680-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><msub><mi>e</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><msub><mi>e</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo stretchy="false">(</mo><msub><mi>e</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><mo stretchy="false">∧</mo><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><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><msub><mi>e</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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-680">\begin{array}[t]{l}
+<div class="inference" id="specialization-attributes-inference"><div class="ruleTitle"><a class="internalDFN" href="#specialization-attributes-inference">Inference 21 (specialization-attributes-inference)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-728-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><msub><mi>e</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><msub><mi>e</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo stretchy="false">(</mo><msub><mi>e</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><mo stretchy="false">∧</mo><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><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><msub><mi>e</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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-728">\begin{array}[t]{l}
 \forall e_1,attrs,e_2.~
 \\
 \qquad entity(e_1,attrs) \wedge specializationOf(e_2,e_1)
@@ -2813,29 +2879,31 @@
 \end{array}</script></div>
 </section>
 <div class="proof">
-  <p> Suppose <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-681-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-681">ent_1 = \rho(e_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-682-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-682">ent_2 = \rho(e_2)</script>.  Suppose
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-683-Frame" style="font-size: 100%; "><span class="MathJax_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><mo stretchy="false">,</mo><mi>v</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-683">(att,v)</script> is an attribute-value pair in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-684-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-684">attrs</script>.  Since
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-685-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>e</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-685">entity(e_1,attrs)</script> holds, we know that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-686-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-686">v \in value(ent_1,att)</script>.
-  Thus <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-687-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-687">v \in value(ent_2,att)</script> since <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-688-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">⊇</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-688">value(ent_2,att) \supseteq value(ent_1,att)</script>.  Since
-  this is the case for all attribute-value pairs in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-689-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-689">attrs</script>, and since
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-690-Frame" style="font-size: 100%; "><span class="MathJax_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-690">e_2</script> obviously denotes an entity, we can conclude <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-691-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>e</mi><mo stretchy="false">,</mo><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-691">W,\rho \models entity(e,attrs</script>).
+  <p> Suppose <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-729-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-729">ent_1 = \rho(e_1)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-730-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>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></math></span></span></span><script type="math/tex" id="MathJax-Element-730">ent_2 = \rho(e_2)</script>.  Suppose
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-731-Frame" style="font-size: 100%; "><span class="MathJax_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><mo stretchy="false">,</mo><mi>v</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-731">(att,v)</script> is an attribute-value pair in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-732-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-732">attrs</script>.  Since
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-733-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>e</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-733">entity(e_1,attrs)</script> holds, we know that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-734-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-734">v \in value(ent_1,att)</script>.
+  Thus <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-735-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><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><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-735">v \in value(ent_2,att)</script> since <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-736-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>n</mi><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">⊇</mo><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mi>n</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-736">value(ent_2,att) \supseteq value(ent_1,att)</script>.  Since
+  this is the case for all attribute-value pairs in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-737-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-737">attrs</script>, and since
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-738-Frame" style="font-size: 100%; "><span class="MathJax_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-738">e_2</script> obviously denotes an entity, we can conclude <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-739-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msub><mi>e</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-739">W,\rho \models entity(e_2,attrs)</script>.
   </p>
 </div>
-
+<div class="todo"> <p> How do we know <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-740-Frame" style="font-size: 100%; "><span class="MathJax_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-740">e_2</script> has
+  generation/invalidation events?  May need axiom.</p>
+  </div>
 <section id="constraints">
 <h3><span class="secno">5.2 </span>Constraints</h3>
 <section id="uniqueness-constraints">
 <h4><span class="secno">5.2.1 </span>Uniqueness constraints</h4>
 
 <div class="constraint" id="key-object"><div class="ruleTitle"><a class="internalDFN" href="#key-object">Constraint 22 (key-object)</a></div>
-  <ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-692-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</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><msub><mi>s</mi><mn>1</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>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><msub><mi>s</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>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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-692">\forall
+  <ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-741-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</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><msub><mi>s</mi><mn>1</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>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><msub><mi>s</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>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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-741">\forall
   id,attrs_1,attrs_2. entity(id,attrs_1) \wedge entity(id,attrs_2)
   \Rightarrow entity(id,attrs_1\cup attrs_2)</script></li>
-  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-693-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msubsup><mi>t</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msubsup><mi>t</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</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>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msubsup><mi>t</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>t</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</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>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">=</mo><msubsup><mi>t</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">=</mo><msubsup><mi>t</mi><mn>2</mn><mo>′</mo></msubsup></math></span></span></span><script type="math/tex" id="MathJax-Element-693">\forall
+  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-742-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msubsup><mi>t</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msubsup><mi>t</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</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>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msubsup><mi>t</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>t</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</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>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">=</mo><msubsup><mi>t</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">=</mo><msubsup><mi>t</mi><mn>2</mn><mo>′</mo></msubsup></math></span></span></span><script type="math/tex" id="MathJax-Element-742">\forall
   id,t_1,t_1',t_2,t_2',attrs_1,attrs_2.~ activity(id,t_1,t_2,attrs_1)
   \wedge activity(id,t_1',t_2',attrs_2) \Rightarrow
   activity(id,t_1,t_2,attrs_1\cup attrs_2) \wedge t_1=t_1' \wedge t_2=t_2'</script></li>
-  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-694-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</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><msub><mi>s</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><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><msub><mi>s</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><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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-694">\forall
+  <li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-743-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</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><msub><mi>s</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><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><msub><mi>s</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><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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-743">\forall
   id,attrs_1,attrs_2. agent(id,attrs_1) \wedge agent(id,attrs_2)
   \Rightarrow agent(id,attrs_1\cup attrs_2)</script>.</li></ol></div>
   <div class="proof">
@@ -2844,7 +2912,7 @@
   for the underlying data.</p>
     </div>
 <div class="constraint" id="key-properties"><div class="ruleTitle"><a class="internalDFN" href="#key-properties">Constraint 23 (key-properties)</a></div>
-  <ol><li><br><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-695-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><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><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>e</mi><mo stretchy="false">=</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>a</mi><mo stretchy="false">=</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>t</mi><mo stretchy="false">=</mo><msup><mi>t</mi><mo>′</mo></msup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-695">\begin{array}[t]{l}
+  <ol><li><br><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-744-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><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><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>e</mi><mo stretchy="false">=</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>a</mi><mo stretchy="false">=</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>t</mi><mo stretchy="false">=</mo><msup><mi>t</mi><mo>′</mo></msup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-744">\begin{array}[t]{l}
     \forall
   id,e,e',a,a',t,t',attrs_1,attrs_2.~ \\
     wasGeneratedBy(id,e,a,t,attrs)
@@ -2852,61 +2920,61 @@
     \quad \Rightarrow
   wasGeneratedBy(id,e,a,t,attrs_1 \cup attrs_2) \wedge e=e' \wedge
   a=a' \wedge t = t'\end{array}</script></li>
-    <li><br> <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-696-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>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><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><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><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><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>e</mi><mo stretchy="false">=</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>a</mi><mo stretchy="false">=</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>t</mi><mo stretchy="false">=</mo><msup><mi>t</mi><mo>′</mo></msup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-696">\begin{array}[t]{l}\forall
+    <li><br> <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-745-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>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><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><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><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><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>e</mi><mo stretchy="false">=</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>a</mi><mo stretchy="false">=</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>t</mi><mo stretchy="false">=</mo><msup><mi>t</mi><mo>′</mo></msup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-745">\begin{array}[t]{l}\forall
   id,e,e',a,a',t,t',attrs_1,attrs_2.~ \\used(id,a,e,t,attrs)
   \wedge used(id,a',e',t',attrs_2)\\ \quad \Rightarrow
   used(id,a,e',t,attrs_1 \cup attrs_2) \wedge e=e' \wedge
   a=a' \wedge t = t'
     \end{array}</script></li>
-    <li><br> <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-697-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</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><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><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><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>1</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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">=</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">=</mo><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-697">\begin{array}[t]{l}\forall
+    <li><br> <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-746-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>o</mi><mi>r</mi><mi>m</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</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><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><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><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>1</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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">=</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">=</mo><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-746">\begin{array}[t]{l}\forall
   id,a_1,a_2,a_1',a_2',attrs_1,attrs_2.~ \\wasInformedBy(id,a_1,a_2,attrs)
   \wedge wasInformedBy(id,a_1',a_2',attrs_2)\\ \Rightarrow
   wasInformedBy(id,a_1,a_2,attrs_1 \cup attrs_2) \wedge a_1=a_1'
   \wedge a_2=a_2'
     \end{array}</script></li>
-    <li><br> <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-698-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>S</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">=</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><mi>e</mi><mo stretchy="false">=</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">=</mo><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><mi>t</mi><mo stretchy="false">=</mo><msup><mi>t</mi><mo>′</mo></msup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-698">\begin{array}[t]{l}\forall
+    <li><br> <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-747-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>S</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">=</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><mi>e</mi><mo stretchy="false">=</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">=</mo><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><mi>t</mi><mo stretchy="false">=</mo><msup><mi>t</mi><mo>′</mo></msup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-747">\begin{array}[t]{l}\forall
   id,e,e'a_1,a_2,a_1',a_2',t,t',attrs_1,attrs_2.~ \\wasStartedBy(id,a_2,e,a_1,t,attrs_1)
   \wedge wasStartedBy(id,a_2',e',a_1',t',attrs_2)\\ \Rightarrow
   wasStartedBy(id,a_2,e,a_1,t,attrs_1 \cup attrs_2) \wedge a_1=a_1'
   \wedge e=e' \wedge
   a_2=a_2' \wedge t = t'
     \end{array}</script></li>
-    <li><br><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-699-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>E</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">=</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><mi>e</mi><mo stretchy="false">=</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">=</mo><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><mi>t</mi><mo stretchy="false">=</mo><msup><mi>t</mi><mo>′</mo></msup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-699">\begin{array}[t]{l}\forall
+    <li><br><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-748-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>E</mi><mi>n</mi><mi>d</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">=</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><mi>e</mi><mo stretchy="false">=</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">=</mo><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><mi>t</mi><mo stretchy="false">=</mo><msup><mi>t</mi><mo>′</mo></msup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-748">\begin{array}[t]{l}\forall
   id,e,e'a_1,a_2,a_1',a_2',t,t',attrs_1,attrs_2.~ \\wasEndedBy(id,a_2,e,a_1,t,attrs_1)
   \wedge wasEndedBy(id,a_2',e',a_1',t',attrs_2)\\ \Rightarrow
   wasEndedBy(id,a_2,e,a_1,t,attrs_1 \cup attrs_2) \wedge a_1=a_1'
   \wedge e=e' \wedge
   a_2=a_2' \wedge t = t'
     \end{array}</script></li>
-    <li><br><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-700-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>e</mi><mo stretchy="false">=</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>a</mi><mo stretchy="false">=</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>t</mi><mo stretchy="false">=</mo><msup><mi>t</mi><mo>′</mo></msup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-700">\begin{array}[t]{l}\forall
+    <li><br><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-749-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>v</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>d</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>e</mi><mo stretchy="false">=</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>a</mi><mo stretchy="false">=</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>t</mi><mo stretchy="false">=</mo><msup><mi>t</mi><mo>′</mo></msup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-749">\begin{array}[t]{l}\forall
   id,e,e',a,a',t,t',attrs_1,attrs_2.~ \\wasInvalidatedBy(id,e,a,t,attrs_1)
   \wedge wasInvalidatedBy(id,e',a',t',attrs_2)\\ \Rightarrow
   wasInvalidatedBy(id,e,a,t,attrs_1 \cup attrs_2) \wedge e=e' \wedge
   a=a' \wedge t = t'
     \end{array}</script></li>
-    <li><br><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-701-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msubsup><mi>e</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msubsup><mi>e</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>g</mi><mo stretchy="false">,</mo><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>u</mi><mo stretchy="false">,</mo><msup><mi>u</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>r</mi><mi>o</mi><mi>m</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>u</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><msubsup><mi>e</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>e</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msubsup><mi>g</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>u</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><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><mo stretchy="false">,</mo><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>u</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">=</mo><msubsup><mi>e</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">=</mo><msubsup><mi>e</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><mi>a</mi><mo stretchy="false">=</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>g</mi><mo stretchy="false">=</mo><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>u</mi><mo stretchy="false">=</mo><msup><mi>u</mi><mo>′</mo></msup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-701">\begin{array}[t]{l}\forall
+    <li><br><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-750-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msubsup><mi>e</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msubsup><mi>e</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>g</mi><mo stretchy="false">,</mo><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>u</mi><mo stretchy="false">,</mo><msup><mi>u</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>r</mi><mi>o</mi><mi>m</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>u</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><msubsup><mi>e</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>e</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><msubsup><mi>g</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>u</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><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><mo stretchy="false">,</mo><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>u</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">=</mo><msubsup><mi>e</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">=</mo><msubsup><mi>e</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><mi>a</mi><mo stretchy="false">=</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>g</mi><mo stretchy="false">=</mo><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>u</mi><mo stretchy="false">=</mo><msup><mi>u</mi><mo>′</mo></msup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-750">\begin{array}[t]{l}\forall
   id,e_1,e_1',e_2,e_2',a,a',g,g',u,u',attrs_1,attrs_2.~ \\wasDerivedFrom(id,e_2,e_1,a,g_2,u_1,attrs_1)
   \wedge wasDerivedFrom(id,e_2',e_1',a',g_2',u_1',attrs_2)\\ \Rightarrow
   wasDerivedFrom(id,e_2,e_1,a,g_2,u_1,attrs_1 \cup attrs_2) \wedge e_1=e_1'\wedge e_2=e_2'\wedge a=a'\wedge g=g'\wedge u=u'
     \end{array}</script></li>
-    <li><br> <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-702-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>T</mi><mi>o</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>e</mi><mo stretchy="false">=</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>a</mi><mi>g</mi><mo stretchy="false">=</mo><mi>a</mi><msup><mi>g</mi><mo>′</mo></msup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-702">\begin{array}[t]{l}\forall
+    <li><br> <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-751-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>T</mi><mi>o</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>e</mi><mo stretchy="false">=</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>a</mi><mi>g</mi><mo stretchy="false">=</mo><mi>a</mi><msup><mi>g</mi><mo>′</mo></msup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-751">\begin{array}[t]{l}\forall
   id,e,e',ag,ag',attrs_1,attrs_2.~ \\wasAttributedTo(id,e,ag,attrs_1)
   \wedge wasAttributedTo(id,e',ag',attrs_2)\\ \Rightarrow
   wasAttributedTo(id,e,ag,attrs_1 \cup attrs_2) \wedge e=e' \wedge ag = ag'
     \end{array}</script></li>
-    <li> <br><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-703-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">,</mo><mi>p</mi><msup><mi>l</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>p</mi><msup><mi>l</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>a</mi><mo stretchy="false">=</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>a</mi><mi>g</mi><mo stretchy="false">=</mo><mi>a</mi><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>p</mi><mi>l</mi><mo stretchy="false">=</mo><mi>p</mi><msup><mi>l</mi><mo>′</mo></msup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-703">\begin{array}[t]{l}\forall
+    <li> <br><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-752-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">,</mo><mi>p</mi><msup><mi>l</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>p</mi><msup><mi>l</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>a</mi><mo stretchy="false">=</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>a</mi><mi>g</mi><mo stretchy="false">=</mo><mi>a</mi><msup><mi>g</mi><mo>′</mo></msup><mo stretchy="false">∧</mo><mi>p</mi><mi>l</mi><mo stretchy="false">=</mo><mi>p</mi><msup><mi>l</mi><mo>′</mo></msup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-752">\begin{array}[t]{l}\forall
   id,a,a',ag,ag',pl,pl',attrs_1,attrs_2.~ \\wasAssociatedWith(id,a,ag,pl,attrs_1)
   \wedge wasAssociatedWith(id,a',ag',pl',attrs_2)\\ \Rightarrow
   wasAssociatedWith(id,a,ag,pl,attrs_1 \cup attrs_2) \wedge a=a'
   \wedge ag=ag'   \wedge pl=pl'
     \end{array}</script></li>
-    <li><br><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-704-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><msubsup><mi>g</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><msubsup><mi>g</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>a</mi><mi>c</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>O</mi><mi>n</mi><mi>B</mi><mi>e</mi><mi>h</mi><mi>a</mi><mi>l</mi><mi>f</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mi>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><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><msubsup><mi>g</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><msubsup><mi>g</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><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><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><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><msubsup><mi>g</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">=</mo><mi>a</mi><msubsup><mi>g</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><mi>a</mi><mo stretchy="false">=</mo><msup><mi>a</mi><mo>′</mo></msup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-704">\begin{array}[t]{l}\forall
+    <li><br><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-753-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><msubsup><mi>g</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><msubsup><mi>g</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>a</mi><mi>c</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>O</mi><mi>n</mi><mi>B</mi><mi>e</mi><mi>h</mi><mi>a</mi><mi>l</mi><mi>f</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mi>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><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><msubsup><mi>g</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><msubsup><mi>g</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msup><mi>a</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><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><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><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><msubsup><mi>g</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">=</mo><mi>a</mi><msubsup><mi>g</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><mi>a</mi><mo stretchy="false">=</mo><msup><mi>a</mi><mo>′</mo></msup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-753">\begin{array}[t]{l}\forall
   id,ag_1,ag_1',ag_2,ag_2',a,a',attrs_1,attrs_2.~ \\actedOnBehalfOf(id,ag_2,ag_1,a,attrs_1)
   \wedge actedOnBehalfOf(id,ag_2',ag_1',a',attrs_2)\\ \Rightarrow
   actedOnBehalfOf(id,ag_2,ag_1,a,attrs_1 \cup attrs_2) \wedge
   ag_1=ag_1' \wedge ag_2 = ag_2' \wedge a = a'
     \end{array}</script></li>
-    <li><br><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-705-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>o</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>o</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msubsup><mi>o</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>o</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msubsup><mi>o</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>o</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msubsup><mi>o</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>o</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>o</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>o</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><msub><mi>o</mi><mn>1</mn></msub><mo stretchy="false">=</mo><msubsup><mi>o</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><msub><mi>o</mi><mn>2</mn></msub><mo stretchy="false">=</mo><msubsup><mi>o</mi><mn>2</mn><mo>′</mo></msubsup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-705">\begin{array}[t]{l}\forall
+    <li><br><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-754-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>o</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>o</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msubsup><mi>o</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>o</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msubsup><mi>o</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>o</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msubsup><mi>o</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msubsup><mi>o</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo stretchy="false">⇒</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>o</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>o</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><msub><mi>o</mi><mn>1</mn></msub><mo stretchy="false">=</mo><msubsup><mi>o</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">∧</mo><msub><mi>o</mi><mn>2</mn></msub><mo stretchy="false">=</mo><msubsup><mi>o</mi><mn>2</mn><mo>′</mo></msubsup></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-754">\begin{array}[t]{l}\forall
   id,o_1,o_2,o_1',o_2',attrs_1,attrs_2.~ \\wasInfluencedBy(id,o_2',o_1',attrs_1)
   \wedge wasInfluencedBy(id,o_2',o_1',attrs_2)\\ \Rightarrow
   wasInfluencedBy(id,o_2,o_1,attrs_1 \cup attrs_2) \wedge o_1=o_1'
@@ -2918,7 +2986,7 @@
   semantics of the respective assertions, again because functions are
   used for the underlying data.</p>
     </div>
-<div class="constraint" id="unique-generation"><div class="ruleTitle"><a class="internalDFN" href="#unique-generation">Constraint 24 (unique-generation)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-706-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-706">\begin{array}[t]{l}
+<div class="constraint" id="unique-generation"><div class="ruleTitle"><a class="internalDFN" href="#unique-generation">Constraint 24 (unique-generation)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-755-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-755">\begin{array}[t]{l}
 \forall gen_1,gen_2,e,a,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasGeneratedBy(gen_1,e,a,t_1,attrs_1) \wedge wasGeneratedBy(gen_2,e,a,t_2,attrs_2)
@@ -2931,7 +2999,7 @@
   This follows from <a href="#axiom17">Axiom 17</a>.
   </p>
   </div>
-<div class="constraint" id="unique-invalidation"><div class="ruleTitle"><a class="internalDFN" href="#unique-invalidation">Constraint 25 (unique-invalidation)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-707-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>n</mi><msub><mi>v</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>n</mi><msub><mi>v</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>1</mn></msub><mo stretchy="false">=</mo><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-707">\begin{array}[t]{l}
+<div class="constraint" id="unique-invalidation"><div class="ruleTitle"><a class="internalDFN" href="#unique-invalidation">Constraint 25 (unique-invalidation)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-756-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>n</mi><msub><mi>v</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>n</mi><msub><mi>v</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>1</mn></msub><mo stretchy="false">=</mo><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-756">\begin{array}[t]{l}
 \forall inv_1,inv_2,e,a,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasInvalidatedBy(inv_1,e,a,t_1,attrs_1) \wedge wasInvalidatedBy(inv_2,e,a,t_2,attrs_2)
@@ -2944,7 +3012,7 @@
   This follows from <a href="#axiom17">Axiom 18</a>.
   </p>
   </div>
-<div class="constraint" id="unique-wasStartedBy"><div class="ruleTitle"><a class="internalDFN" href="#unique-wasStartedBy">Constraint 26 (unique-wasStartedBy)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-708-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>2</mn></msub><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>0</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><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>0</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>2</mn></msub><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>0</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">=</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-708">\begin{array}[t]{l}
+<div class="constraint" id="unique-wasStartedBy"><div class="ruleTitle"><a class="internalDFN" href="#unique-wasStartedBy">Constraint 26 (unique-wasStartedBy)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-757-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>2</mn></msub><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>0</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><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>0</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>2</mn></msub><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>0</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">=</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-757">\begin{array}[t]{l}
 \forall start_1,start_2,a,e_1,e_2,a_0,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasStartedBy(start_1,a,e_1,a_0,t_1,attrs_1) \wedge wasStartedBy(start_2,a,e_2,a_0,t_2,attrs_2)
@@ -2958,7 +3026,7 @@
   </p>
   </div>
 
-<div class="constraint" id="unique-wasEndedBy"><div class="ruleTitle"><a class="internalDFN" href="#unique-wasEndedBy">Constraint 27 (unique-wasEndedBy)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-709-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>0</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>e</mi><mi>n</mi><msub><mi>d</mi><mn>1</mn></msub><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>0</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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>0</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>1</mn></msub><mo stretchy="false">=</mo><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-709">\begin{array}[t]{l}
+<div class="constraint" id="unique-wasEndedBy"><div class="ruleTitle"><a class="internalDFN" href="#unique-wasEndedBy">Constraint 27 (unique-wasEndedBy)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-758-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>0</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>e</mi><mi>n</mi><msub><mi>d</mi><mn>1</mn></msub><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>0</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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>0</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>1</mn></msub><mo stretchy="false">=</mo><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-758">\begin{array}[t]{l}
 \forall end_1,end_2,a,e_1,e_2,a_0,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasEndedBy(end_1,a,e_1,a_0,t_1,attrs_1) \wedge wasEndedBy(end_2,a,e_2,a_0,t_2,attrs_2)
@@ -2971,7 +3039,7 @@
   This follows from <a href="#axiom17">Axiom 20</a>.
   </p>
   </div>
-<div class="constraint" id="unique-startTime"><div class="ruleTitle"><a class="internalDFN" href="#unique-startTime">Constraint 28 (unique-startTime)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-710-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">=</mo><mi>t</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-710">\begin{array}[t]{l}
+<div class="constraint" id="unique-startTime"><div class="ruleTitle"><a class="internalDFN" href="#unique-startTime">Constraint 28 (unique-startTime)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-759-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">=</mo><mi>t</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-759">\begin{array}[t]{l}
 \forall start,a_1,a_2,t,t_1,t_2,e,attrs,attrs_1.~
 \\
 \qquad activity(a_2,t_1,t_2,attrs) \wedge wasStartedBy(start,a_2,e,a_1,t,attrs_1)
@@ -2981,11 +3049,11 @@
 \end{array}</script></div>
 <div class="proof">
   <p>
-  This follows from the semantics of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-711-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-711">wasStartedBy</script>, since the start times
+  This follows from the semantics of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-760-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-760">wasStartedBy</script>, since the start times
   must both match that of the activity.
   </p>
   </div>
-<div class="constraint" id="unique-endTime"><div class="ruleTitle"><a class="internalDFN" href="#unique-endTime">Constraint 29 (unique-endTime)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-712-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><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>e</mi><mi>n</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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">=</mo><mi>t</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-712">\begin{array}[t]{l}
+<div class="constraint" id="unique-endTime"><div class="ruleTitle"><a class="internalDFN" href="#unique-endTime">Constraint 29 (unique-endTime)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-761-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><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>e</mi><mi>n</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><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">=</mo><mi>t</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-761">\begin{array}[t]{l}
 \forall end,a_1,a_2,t,t_1,t_2,e,attrs,attrs_1.~
 \\
 \qquad activity(a_2,t_1,t_2,attrs) \wedge wasEndedBy(end,a_2,e,a_1,t,attrs_1)
@@ -2995,7 +3063,7 @@
 \end{array}</script></div>
 <div class="proof">
   <p>
-  This follows from the semantics of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-713-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-713">wasEndedBy</script>, since the end times
+  This follows from the semantics of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-762-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-762">wasEndedBy</script>, since the end times
   must both match that of the activity.
   </p>
   </div>
@@ -3005,7 +3073,7 @@
 <h4><span class="secno">5.2.2 </span>Ordering constraints</h4>
 
 
-<div class="constraint" id="start-precedes-end"><div class="ruleTitle"><a class="internalDFN" href="#start-precedes-end">Constraint 30 (start-precedes-end)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-714-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><mi>d</mi><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><mi>d</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-714">\begin{array}[t]{l}
+<div class="constraint" id="start-precedes-end"><div class="ruleTitle"><a class="internalDFN" href="#start-precedes-end">Constraint 30 (start-precedes-end)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-763-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><mi>d</mi><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><mi>d</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-763">\begin{array}[t]{l}
 \forall start,end,a,e_1,e_2,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasStartedBy(start,a,e_1,a_1,t_1,attrs_1) \wedge wasEndedBy(end,a,e_2,a_2,t_2,attrs_2)
@@ -3018,7 +3086,7 @@
   </p>
   </div>
   
-<div class="constraint" id="start-start-ordering"><div class="ruleTitle"><a class="internalDFN" href="#start-start-ordering">Constraint 31 (start-start-ordering)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-715-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>2</mn></msub><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>2</mn></msub><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-715">\begin{array}[t]{l}
+<div class="constraint" id="start-start-ordering"><div class="ruleTitle"><a class="internalDFN" href="#start-start-ordering">Constraint 31 (start-start-ordering)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-764-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>2</mn></msub><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>2</mn></msub><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-764">\begin{array}[t]{l}
 \forall start_1,start_2,a,e_1,e_2,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasStartedBy(start_1,a,e_1,a_1,t_1,attrs_1) \wedge wasStartedBy(start_2,a,e_2,a_2,t_2,attrs_2)
@@ -3031,7 +3099,7 @@
   </p>
   </div>
   
-<div class="constraint" id="end-end-ordering"><div class="ruleTitle"><a class="internalDFN" href="#end-end-ordering">Constraint 32 (end-end-ordering)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-716-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>e</mi><mi>n</mi><msub><mi>d</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-716">\begin{array}[t]{l}
+<div class="constraint" id="end-end-ordering"><div class="ruleTitle"><a class="internalDFN" href="#end-end-ordering">Constraint 32 (end-end-ordering)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-765-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>e</mi><mi>n</mi><msub><mi>d</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-765">\begin{array}[t]{l}
 \forall end_1,end_2,a,e_1,e_2,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasEndedBy(end_1,a,e_1,a_1,t_1,attrs_1) \wedge wasEndedBy(end_2,a,e_2,a_2,t_2,attrs_2)
@@ -3044,14 +3112,14 @@
   </p>
   </div>
   
-<div class="constraint" id="usage-within-activity"><div class="ruleTitle"><a class="internalDFN" href="#usage-within-activity">Constraint 33 (usage-within-activity)</a></div><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-717-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">,</mo><mi>u</mi><mi>s</mi><mi>e</mi><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><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><mi>a</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>u</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-717">\begin{array}[t]{l}
+<div class="constraint" id="usage-within-activity"><div class="ruleTitle"><a class="internalDFN" href="#usage-within-activity">Constraint 33 (usage-within-activity)</a></div><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-766-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">,</mo><mi>u</mi><mi>s</mi><mi>e</mi><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><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><mi>a</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>u</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-766">\begin{array}[t]{l}
 \forall start,use,a,e_1,e_2,a_1,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasStartedBy(start,a,e_1,a_1,t_1,attrs_1) \wedge used(use,a,e_2,t_2,attrs_2)
 \\
 \quad\Rightarrow
 start \precedes use
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-718-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>u</mi><mi>s</mi><mi>e</mi><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><mi>d</mi><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><mi>a</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><mi>d</mi><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>u</mi><mi>s</mi><mi>e</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><mi>d</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-718">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-767-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>u</mi><mi>s</mi><mi>e</mi><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><mi>d</mi><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><mi>a</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><mi>d</mi><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>u</mi><mi>s</mi><mi>e</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><mi>d</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-767">\begin{array}[t]{l}
 \forall use,end,a,e_1,e_2,a_2,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad used(use,a,e_1,t_1,attrs_1) \wedge wasEndedBy(end,a,e_2,a_2,t_2,attrs_2)
@@ -3065,14 +3133,14 @@
   </p>
   </div>
   
-<div class="constraint" id="generation-within-activity"><div class="ruleTitle"><a class="internalDFN" href="#generation-within-activity">Constraint 34 (generation-within-activity)</a></div><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-719-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>g</mi><mi>e</mi><mi>n</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-719">\begin{array}[t]{l}
+<div class="constraint" id="generation-within-activity"><div class="ruleTitle"><a class="internalDFN" href="#generation-within-activity">Constraint 34 (generation-within-activity)</a></div><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-768-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>g</mi><mi>e</mi><mi>n</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-768">\begin{array}[t]{l}
 \forall start,gen,e_1,e_2,a,a_1,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasStartedBy(start,a,e_1,a_1,t_1,attrs_1) \wedge wasGeneratedBy(gen,e_2,a,t_2,attrs_2)
 \\
 \quad\Rightarrow
 start \precedes gen
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-720-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><msub><mi>t</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><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>g</mi><mi>e</mi><mi>n</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><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>e</mi><mi>n</mi><mi>d</mi><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><mi>n</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><mi>d</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-720">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-769-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><msub><mi>t</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><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>g</mi><mi>e</mi><mi>n</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><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>e</mi><mi>n</mi><mi>d</mi><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><mi>n</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><mi>d</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-769">\begin{array}[t]{l}
 \forall gen,end,e,e_1,a,a_1,t,t_1,attrs,attrs_1.~
 \\
 \qquad wasGeneratedBy(gen,e,a,t,attrs) \wedge wasEndedBy(end,a,e_1,a_1,t_1,attrs_1)
@@ -3086,7 +3154,7 @@
   </p>
   </div>
   
-<div class="constraint" id="wasInformedBy-ordering"><div class="ruleTitle"><a class="internalDFN" href="#wasInformedBy-ordering">Constraint 35 (wasInformedBy-ordering)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-721-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</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><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</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><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><mi>d</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-721">\begin{array}[t]{l}
+<div class="constraint" id="wasInformedBy-ordering"><div class="ruleTitle"><a class="internalDFN" href="#wasInformedBy-ordering">Constraint 35 (wasInformedBy-ordering)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-770-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</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><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</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><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>1</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msubsup><mi>a</mi><mn>2</mn><mo>′</mo></msubsup><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><mi>d</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-770">\begin{array}[t]{l}
 \forall id,start,end,a_1,a_1',a_2,a_2',e_1,e_2,t_1,t_2,attrs,attrs_1,attrs_2.~
 \\
 \qquad wasInformedBy(id,a_2,a_1,attrs) \wedge wasStartedBy(start,a_1,e_1,a_1',t_1,attrs_1) \wedge wasEndedBy(end,a_2,e_2,a_2',t_2,attrs_2)
@@ -3095,14 +3163,14 @@
 start \precedes end
 \end{array}</script></div>
 <div class="proof">
-  <p>This follows from the semantics of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-722-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-722">wasInformedBy</script> and the
-  previous two constraints, because <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-723-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-723">wasInformedBy</script> implies the
-  existence of intermediate generation and usage events linking <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-724-Frame" style="font-size: 100%; "><span class="MathJax_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-724">a_1</script>
-  and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-725-Frame" style="font-size: 100%; "><span class="MathJax_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-725">a_2</script>.
+  <p>This follows from the semantics of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-771-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-771">wasInformedBy</script> and the
+  previous two constraints, because <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-772-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-772">wasInformedBy</script> implies the
+  existence of intermediate generation and usage events linking <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-773-Frame" style="font-size: 100%; "><span class="MathJax_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-773">a_1</script>
+  and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-774-Frame" style="font-size: 100%; "><span class="MathJax_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-774">a_2</script>.
   </p>
   </div>
   
-<div class="constraint" id="generation-precedes-invalidation"><div class="ruleTitle"><a class="internalDFN" href="#generation-precedes-invalidation">Constraint 36 (generation-precedes-invalidation)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-726-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>n</mi><mi>v</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><mi>n</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>i</mi><mi>n</mi><mi>v</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-726">\begin{array}[t]{l}
+<div class="constraint" id="generation-precedes-invalidation"><div class="ruleTitle"><a class="internalDFN" href="#generation-precedes-invalidation">Constraint 36 (generation-precedes-invalidation)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-775-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>n</mi><mi>v</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><mi>n</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>i</mi><mi>n</mi><mi>v</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-775">\begin{array}[t]{l}
 \forall gen,inv,e,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasGeneratedBy(gen,e,a_1,t_1,attrs_1) \wedge wasInvalidatedBy(inv,e,a_2,t_2,attrs_2)
@@ -3116,7 +3184,7 @@
   </p>
   </div>
 
-  <div class="constraint" id="generation-precedes-usage"><div class="ruleTitle"><a class="internalDFN" href="#generation-precedes-usage">Constraint 37 (generation-precedes-usage)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-727-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><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><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><mi>n</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>u</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-727">\begin{array}[t]{l}
+  <div class="constraint" id="generation-precedes-usage"><div class="ruleTitle"><a class="internalDFN" href="#generation-precedes-usage">Constraint 37 (generation-precedes-usage)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-776-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><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><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><mi>n</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>u</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-776">\begin{array}[t]{l}
 \forall gen,use,e,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasGeneratedBy(gen,e,a_1,t_1,attrs_1) \wedge used(use,a_2,e,t_2,attrs_2)
@@ -3130,7 +3198,7 @@
   </p>
   </div>
 
-<div class="constraint" id="usage-precedes-invalidation"><div class="ruleTitle"><a class="internalDFN" href="#usage-precedes-invalidation">Constraint 38 (usage-precedes-invalidation)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-728-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>u</mi><mi>s</mi><mi>e</mi><mo stretchy="false">,</mo><mi>i</mi><mi>n</mi><mi>v</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><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>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>n</mi><mi>v</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>u</mi><mi>s</mi><mi>e</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>i</mi><mi>n</mi><mi>v</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-728">\begin{array}[t]{l}
+<div class="constraint" id="usage-precedes-invalidation"><div class="ruleTitle"><a class="internalDFN" href="#usage-precedes-invalidation">Constraint 38 (usage-precedes-invalidation)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-777-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>u</mi><mi>s</mi><mi>e</mi><mo stretchy="false">,</mo><mi>i</mi><mi>n</mi><mi>v</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><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>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>n</mi><mi>v</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>u</mi><mi>s</mi><mi>e</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>i</mi><mi>n</mi><mi>v</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-777">\begin{array}[t]{l}
 \forall use,inv,a_1,a_2,e,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad used(use,a_1,e,t_1,attrs_1) \wedge wasInvalidatedBy(inv,e,a_2,t_2,attrs_2)
@@ -3144,7 +3212,7 @@
   </p>
   </div>
 
-<div class="constraint" id="generation-generation-ordering"><div class="ruleTitle"><a class="internalDFN" href="#generation-generation-ordering">Constraint 39 (generation-generation-ordering)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-729-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-729">\begin{array}[t]{l}
+<div class="constraint" id="generation-generation-ordering"><div class="ruleTitle"><a class="internalDFN" href="#generation-generation-ordering">Constraint 39 (generation-generation-ordering)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-778-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-778">\begin{array}[t]{l}
 \forall gen_1,gen_2,e,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasGeneratedBy(gen_1,e,a_1,t_1,attrs_1) \wedge wasGeneratedBy(gen_2,e,a_2,t_2,attrs_2)
@@ -3158,7 +3226,7 @@
   </p>
   </div>
 
-<div class="constraint" id="invalidation-invalidation-ordering"><div class="ruleTitle"><a class="internalDFN" href="#invalidation-invalidation-ordering">Constraint 40 (invalidation-invalidation-ordering)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-730-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>i</mi><mi>n</mi><msub><mi>v</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><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>n</mi><msub><mi>v</mi><mn>1</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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>n</mi><msub><mi>v</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-730">\begin{array}[t]{l}
+<div class="constraint" id="invalidation-invalidation-ordering"><div class="ruleTitle"><a class="internalDFN" href="#invalidation-invalidation-ordering">Constraint 40 (invalidation-invalidation-ordering)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-779-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>i</mi><mi>n</mi><msub><mi>v</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><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>n</mi><msub><mi>v</mi><mn>1</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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>n</mi><msub><mi>v</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-779">\begin{array}[t]{l}
 \forall inv_1,inv_2,e,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasInvalidatedBy(inv_1,e,a_1,t_1,attrs_1) \wedge wasInvalidatedBy(inv_2,e,a_2,t_2,attrs_2)
@@ -3172,7 +3240,7 @@
   </p>
   </div>
 
-<div class="constraint" id="derivation-usage-generation-ordering"><div class="ruleTitle"><a class="internalDFN" href="#derivation-usage-generation-ordering">Constraint 41 (derivation-usage-generation-ordering)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-731-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>u</mi><mi>s</mi><msub><mi>e</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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>u</mi><mi>s</mi><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>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><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>u</mi><mi>s</mi><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>u</mi><mi>s</mi><msub><mi>e</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-731">\begin{array}[t]{l}
+<div class="constraint" id="derivation-usage-generation-ordering"><div class="ruleTitle"><a class="internalDFN" href="#derivation-usage-generation-ordering">Constraint 41 (derivation-usage-generation-ordering)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-780-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>u</mi><mi>s</mi><msub><mi>e</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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>u</mi><mi>s</mi><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>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><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>u</mi><mi>s</mi><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>u</mi><mi>s</mi><msub><mi>e</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-780">\begin{array}[t]{l}
 \forall d,e_1,e_2,a,gen_2,use_1,attrs.~
 \\
 \qquad notNull(a) \wedge notNull(gen_2) \wedge notNull(use_1) \wedge wasDerivedFrom(d,e_2,e_1,a,gen_2,use_1,attrs)
@@ -3186,7 +3254,7 @@
   </p>
   </div>
 
-<div class="constraint" id="derivation-generation-generation-ordering"><div class="ruleTitle"><a class="internalDFN" href="#derivation-generation-generation-ordering">Constraint 42 (derivation-generation-generation-ordering)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-732-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</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><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>g</mi><mo stretchy="false">,</mo><mi>u</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>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><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>g</mi><mi>e</mi><msub><mi>n</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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>g</mi><mi>e</mi><msub><mi>n</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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">strictlyPrecedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-732">\begin{array}[t]{l}
+<div class="constraint" id="derivation-generation-generation-ordering"><div class="ruleTitle"><a class="internalDFN" href="#derivation-generation-generation-ordering">Constraint 42 (derivation-generation-generation-ordering)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-781-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</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><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>g</mi><mo stretchy="false">,</mo><mi>u</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>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><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>g</mi><mi>e</mi><msub><mi>n</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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>g</mi><mi>e</mi><msub><mi>n</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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">strictlyPrecedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-781">\begin{array}[t]{l}
 \forall d,gen_1,gen_2,e_1,e_2,a,a_1,a_2,g,u,t_1,t_2,attrs,attrs_1,attrs_2.~
 \\
 \qquad wasDerivedFrom(d,e_2,e_1,a,g,u,attrs) \wedge wasGeneratedBy(gen_1,e_1,a_1,t_1,attrs_1) \wedge wasGeneratedBy(gen_2,e_2,a_2,t_2,attrs_2)
@@ -3200,14 +3268,14 @@
   </p>
   </div>
 
-<div class="constraint" id="wasStartedBy-ordering"><div class="ruleTitle"><a class="internalDFN" href="#wasStartedBy-ordering">Constraint 43 (wasStartedBy-ordering)</a></div><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-733-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><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><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><mi>n</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-733">\begin{array}[t]{l}
+<div class="constraint" id="wasStartedBy-ordering"><div class="ruleTitle"><a class="internalDFN" href="#wasStartedBy-ordering">Constraint 43 (wasStartedBy-ordering)</a></div><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-782-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><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><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><mi>n</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-782">\begin{array}[t]{l}
 \forall gen,start,e,a,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasGeneratedBy(gen,e,a_1,t_1,attrs_1) \wedge wasStartedBy(start,a,e,a_2,t_2,attrs_2)
 \\
 \quad\Rightarrow
 gen \precedes start
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-734-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>n</mi><mi>v</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>i</mi><mi>n</mi><mi>v</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-734">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-783-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>n</mi><mi>v</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>i</mi><mi>n</mi><mi>v</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-783">\begin{array}[t]{l}
 \forall start,inv,e,a,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasStartedBy(start,a,e,a_1,t_1,attrs_1) \wedge wasInvalidatedBy(inv,e,a_2,t_2,attrs_2)
@@ -3222,14 +3290,14 @@
   </p>
   </div>
 
-<div class="constraint" id="wasEndedBy-ordering"><div class="ruleTitle"><a class="internalDFN" href="#wasEndedBy-ordering">Constraint 44 (wasEndedBy-ordering)</a></div><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-735-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><mi>n</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><mi>d</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-735">\begin{array}[t]{l}
+<div class="constraint" id="wasEndedBy-ordering"><div class="ruleTitle"><a class="internalDFN" href="#wasEndedBy-ordering">Constraint 44 (wasEndedBy-ordering)</a></div><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-784-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>g</mi><mi>e</mi><mi>n</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><mi>n</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><mi>d</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-784">\begin{array}[t]{l}
 \forall gen,end,e,a,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasGeneratedBy(gen,e,a_1,t_1,attrs_1) \wedge wasEndedBy(end,a,e,a_2,t_2,attrs_2)
 \\
 \quad\Rightarrow
 gen \precedes end
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-736-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><mi>i</mi><mi>n</mi><mi>v</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>n</mi><mi>v</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>e</mi><mi>n</mi><mi>d</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>i</mi><mi>n</mi><mi>v</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-736">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-785-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><mi>i</mi><mi>n</mi><mi>v</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>n</mi><mi>v</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>e</mi><mi>n</mi><mi>d</mi><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>i</mi><mi>n</mi><mi>v</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-785">\begin{array}[t]{l}
 \forall end,inv,e,a,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasEndedBy(end,a,e,a_1,t_1,attrs_1) \wedge wasInvalidatedBy(inv,e,a_2,t_2,attrs_2)
@@ -3244,7 +3312,7 @@
   </p>
   </div>
 
-<div class="constraint" id="specialization-generation-ordering"><div class="ruleTitle"><a class="internalDFN" href="#specialization-generation-ordering">Constraint 45 (specialization-generation-ordering)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-737-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>g</mi><mi>e</mi><msub><mi>n</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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>g</mi><mi>e</mi><msub><mi>n</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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-737">\begin{array}[t]{l}
+<div class="constraint" id="specialization-generation-ordering"><div class="ruleTitle"><a class="internalDFN" href="#specialization-generation-ordering">Constraint 45 (specialization-generation-ordering)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-786-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>g</mi><mi>e</mi><msub><mi>n</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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>g</mi><mi>e</mi><msub><mi>n</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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-786">\begin{array}[t]{l}
 \forall gen_1,gen_2,e_1,e_2,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad specializationOf(e_2,e_1) \wedge wasGeneratedBy(gen_1,e_1,a_1,t_1,attrs_1) \wedge wasGeneratedBy(gen_2,e_2,a_2,t_2,attrs_2)
@@ -3255,11 +3323,14 @@
 
 
 <div class="proof">
-  <p> ???
+  <p> This follows from <a href="#axiom23">Axiom 23</a> and the fact
+  that if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-787-Frame" style="font-size: 100%; "><span class="MathJax_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-787">e_2</script> specializes <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-788-Frame" style="font-size: 100%; "><span class="MathJax_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-788">e_1</script> then all of the events of the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-789-Frame" style="font-size: 100%; "><span class="MathJax_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-789">e_2</script>
+  are events of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-790-Frame" style="font-size: 100%; "><span class="MathJax_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-790">e_1</script>.  Thus, the generation of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-791-Frame" style="font-size: 100%; "><span class="MathJax_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-791">e_1</script> precedes all
+  events of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-792-Frame" style="font-size: 100%; "><span class="MathJax_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-792">e_2</script>.
   </p>
   </div>
 
-  <div class="constraint" id="specialization-invalidation-ordering"><div class="ruleTitle"><a class="internalDFN" href="#specialization-invalidation-ordering">Constraint 46 (specialization-invalidation-ordering)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-738-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo><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>n</mi><msub><mi>v</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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>n</mi><msub><mi>v</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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-738">\begin{array}[t]{l}
+  <div class="constraint" id="specialization-invalidation-ordering"><div class="ruleTitle"><a class="internalDFN" href="#specialization-invalidation-ordering">Constraint 46 (specialization-invalidation-ordering)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-793-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo><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>n</mi><msub><mi>v</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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>n</mi><msub><mi>v</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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-793">\begin{array}[t]{l}
 \forall inv_1,inv_2,e_1,e_2,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad specializationOf(e_1,e_2) \wedge wasInvalidatedBy(inv_1,e_1,a_1,t_1,attrs_1) \wedge wasInvalidatedBy(inv_2,e_2,a_2,t_2,attrs_2)
@@ -3269,32 +3340,35 @@
 \end{array}</script></div>
 
 <div class="proof">
-  <p> ???
+  <p> This follows from <a href="#axiom23">Axiom 24</a> and the fact
+  that if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-794-Frame" style="font-size: 100%; "><span class="MathJax_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-794">e_2</script> specializes <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-795-Frame" style="font-size: 100%; "><span class="MathJax_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-795">e_1</script> then all of the events of the <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-796-Frame" style="font-size: 100%; "><span class="MathJax_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-796">e_2</script>
+  are events of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-797-Frame" style="font-size: 100%; "><span class="MathJax_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-797">e_1</script>.  Thus, the invalidation of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-798-Frame" style="font-size: 100%; "><span class="MathJax_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-798">e_1</script> follows all
+  events of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-799-Frame" style="font-size: 100%; "><span class="MathJax_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-799">e_2</script>.
   </p>
   </div>
 
-<div class="constraint" id="wasAssociatedWith-ordering"><div class="ruleTitle"><a class="internalDFN" href="#wasAssociatedWith-ordering">Constraint 47 (wasAssociatedWith-ordering)</a></div><p>In the following inferences, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-739-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-739">pl</script> may be a placeholder -.</p><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-740-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mo stretchy="false">,</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>2</mn></msub><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</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><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>n</mi><msub><mi>v</mi><mn>2</mn></msub><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-740">\begin{array}[t]{l}
+<div class="constraint" id="wasAssociatedWith-ordering"><div class="ruleTitle"><a class="internalDFN" href="#wasAssociatedWith-ordering">Constraint 47 (wasAssociatedWith-ordering)</a></div><p>In the following inferences, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-800-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-800">pl</script> may be a placeholder -.</p><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-801-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mo stretchy="false">,</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>2</mn></msub><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</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><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>n</mi><msub><mi>v</mi><mn>2</mn></msub><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-801">\begin{array}[t]{l}
 \forall assoc,start_1,inv_2,ag,e_1,e_2,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasAssociatedWith(assoc,a,ag,pl,attrs) \wedge wasStartedBy(start_1,a,e_1,a_1,t_1,attrs_1) \wedge wasInvalidatedBy(inv_2,ag,a_2,t_2,attrs_2)
 \\
 \quad\Rightarrow
 start_1 \precedes inv_2
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-741-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mo stretchy="false">,</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</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><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-741">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-802-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mo stretchy="false">,</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</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><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-802">\begin{array}[t]{l}
 \forall assoc,gen_1,end_2,ag,e_1,e_2,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasAssociatedWith(assoc,a,ag,pl,attrs) \wedge wasGeneratedBy(gen_1,ag,a_1,t_1,attrs_1) \wedge wasEndedBy(end_2,a,e_2,a_2,t_2,attrs_2)
 \\
 \quad\Rightarrow
 gen_1 \precedes end_2
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-742-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mo stretchy="false">,</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</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><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-742">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-803-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mo stretchy="false">,</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</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><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-803">\begin{array}[t]{l}
 \forall assoc,start_1,end_2,ag,e_1,e_2,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasAssociatedWith(assoc,a,ag,pl,attrs) \wedge wasStartedBy(start_1,a,e_1,a_1,t_1,attrs_1) \wedge wasEndedBy(end_2,ag,e_2,a_2,t_2,attrs_2)
 \\
 \quad\Rightarrow
 start_1 \precedes end_2
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-743-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mo stretchy="false">,</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</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><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-743">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-804-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mo stretchy="false">,</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>a</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</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><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-804">\begin{array}[t]{l}
 \forall assoc,start_1,end_2,ag,e_1,e_2,a_1,a_2,t_1,t_2,attrs_1,attrs_2.~
 \\
 \qquad wasAssociatedWith(assoc,a,ag,pl,attrs) \wedge wasStartedBy(start_1,ag,e_1,a_1,t_1,attrs_1) \wedge wasEndedBy(end_2,a,e_2,a_2,t_2,attrs_2)
@@ -3304,18 +3378,19 @@
 \end{array}</script></li></ol></div>
 
 <div class="proof">
-  <p>TODO - arbitrary axioms.
+  <p>The four parts follow from <a href="#axiom27">Axiom 27</a> through
+  <a href="#axiom30">Axiom 30</a> respectively.
   </p>
   </div>
 
-<div class="constraint" id="wasAttributedTo-ordering"><div class="ruleTitle"><a class="internalDFN" href="#wasAttributedTo-ordering">Constraint 48 (wasAttributedTo-ordering)</a></div><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-744-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>a</mi><mi>t</mi><mi>t</mi><mo stretchy="false">,</mo><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><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><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><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>a</mi><mi>t</mi><mi>t</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><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-744">\begin{array}[t]{l}
+<div class="constraint" id="wasAttributedTo-ordering"><div class="ruleTitle"><a class="internalDFN" href="#wasAttributedTo-ordering">Constraint 48 (wasAttributedTo-ordering)</a></div><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-805-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>a</mi><mi>t</mi><mi>t</mi><mo stretchy="false">,</mo><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><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><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><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>a</mi><mi>t</mi><mi>t</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><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-805">\begin{array}[t]{l}
 \forall att,gen_1,gen_2,e,a_1,a_2,t_1,t_2,ag,attrs,attrs_1,attrs_2.~
 \\
 \qquad wasAttributedTo(att,e,ag,attrs) \wedge wasGeneratedBy(gen_1,ag,a_1,t_1,attrs_1) \wedge wasGeneratedBy(gen_2,e,a_2,t_2,attrs_2)
 \\
 \quad\Rightarrow
 gen_1 \precedes gen_2
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-745-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>a</mi><mi>t</mi><mi>t</mi><mo stretchy="false">,</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</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><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>a</mi><mi>t</mi><mi>t</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><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-745">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-806-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>a</mi><mi>t</mi><mi>t</mi><mo stretchy="false">,</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</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><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>a</mi><mi>t</mi><mi>t</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><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-806">\begin{array}[t]{l}
 \forall att,start_1,gen_2,e,e_1,a_1,a_2,ag,t_1,t_2,attrs,attrs_1,attrs_2.~
 \\
 \qquad wasAttributedTo(att,e,ag,attrs) \wedge wasStartedBy(start_1,ag,e_1,a_1,t_1,attrs_1) \wedge wasGeneratedBy(gen_2,e,a_2,t_2,attrs_2)
@@ -3325,18 +3400,19 @@
 \end{array}</script></li></ol></div>
 
 <div class="proof">
-  <p>TODO - arbitrary axioms.
+  <p>These properties follow from <a href="#axiom31">Axiom 31</a> and
+  <a href="#axiom 32">Axiom 32</a>.
   </p>
   </div>
 
-<div class="constraint" id="actedOnBehalfOf-ordering"><div class="ruleTitle"><a class="internalDFN" href="#actedOnBehalfOf-ordering">Constraint 49 (actedOnBehalfOf-ordering)</a></div><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-746-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>d</mi><mi>e</mi><mi>l</mi><mo stretchy="false">,</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>i</mi><mi>n</mi><msub><mi>v</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><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>d</mi><mi>e</mi><mi>l</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><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>n</mi><msub><mi>v</mi><mn>2</mn></msub><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-746">\begin{array}[t]{l}
+<div class="constraint" id="actedOnBehalfOf-ordering"><div class="ruleTitle"><a class="internalDFN" href="#actedOnBehalfOf-ordering">Constraint 49 (actedOnBehalfOf-ordering)</a></div><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-807-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>d</mi><mi>e</mi><mi>l</mi><mo stretchy="false">,</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>i</mi><mi>n</mi><msub><mi>v</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><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>d</mi><mi>e</mi><mi>l</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><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><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>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>n</mi><msub><mi>v</mi><mn>2</mn></msub><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>g</mi><mi>e</mi><msub><mi>n</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>i</mi><mi>n</mi><msub><mi>v</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-807">\begin{array}[t]{l}
 \forall del,gen_1,inv_2,ag_1,ag_2,a,a_1,a_2,t_1,t_2,attrs,attrs_1,attrs_2.~
 \\
 \qquad actedOnBehalfOf(del,ag_2,ag_1,a,attrs) \wedge wasGeneratedBy(gen_1,ag_1,a_1,t_1,attrs_1) \wedge wasInvalidatedBy(inv_2,ag_2,a_2,t_2,attrs_2)
 \\
 \quad\Rightarrow
 gen_1 \precedes inv_2
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-747-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>d</mi><mi>e</mi><mi>l</mi><mo stretchy="false">,</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><msub><mi>d</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><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</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><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>d</mi><mi>e</mi><mi>l</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><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-747">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-808-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>d</mi><mi>e</mi><mi>l</mi><mo stretchy="false">,</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><msub><mi>d</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><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</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><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>d</mi><mi>e</mi><mi>l</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><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><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><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub><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><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><msub><mi>t</mi><mn>1</mn></msub><mtext>&nbsp;</mtext><mrow class="MJX-TeXAtom-REL"><mrow class="MJX-TeXAtom-ORD"><mtext mathvariant="normal">precedes</mtext></mrow></mrow><mtext>&nbsp;</mtext><mi>e</mi><mi>n</mi><msub><mi>d</mi><mn>2</mn></msub></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-808">\begin{array}[t]{l}
 \forall del,start_1,end_2,ag_1,ag_2,a,a_1,a_2,e_1,e_2,t_1,t_2,attrs,attrs_1,attrs_2.~
 \\
 \qquad actedOnBehalfOf(del,ag_2,ag_1,a,attrs) \wedge wasStartedBy(start_1,ag_1,e_1,a_1,t_1,attrs_1) \wedge wasEndedBy(end_2,ag_2,e_2,a_2,t_2,attrs_2)
@@ -3346,7 +3422,8 @@
 \end{array}</script></li></ol></div>
 
 <div class="proof">
-  <p>TODO - arbitrary axioms.
+  <p>These properties follow from <a href="#axiom33">Axiom 33</a> and
+  <a href="#axiom 34">Axiom 34</a>.
   </p>
   </div>
 
@@ -3356,133 +3433,133 @@
 <section id="typing-constraints">
 <h4><span class="secno">5.2.3 </span>Typing constraints</h4>
 
-<div class="constraint" id="typing"><div class="ruleTitle"><a class="internalDFN" href="#typing">Constraint 50 (typing)</a></div><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-748-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-748">\begin{array}[t]{l}
+<div class="constraint" id="typing"><div class="ruleTitle"><a class="internalDFN" href="#typing">Constraint 50 (typing)</a></div><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-809-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-809">\begin{array}[t]{l}
 \forall e,attrs.~
 \\
 \qquad entity(e,attrs)
 \\
 \quad\Rightarrow
 typeOf(e,entity)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-749-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-749">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-810-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-810">\begin{array}[t]{l}
 \forall ag,attrs.~
 \\
 \qquad agent(ag,attrs)
 \\
 \quad\Rightarrow
 typeOf(ag,agent)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-750-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><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>y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-750">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-811-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><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>y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-811">\begin{array}[t]{l}
 \forall a,t_1,t_2,attrs.~
 \\
 \qquad activity(a,t_1,t_2,attrs)
 \\
 \quad\Rightarrow
 typeOf(a,activity)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-751-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>u</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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>u</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></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><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>y</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-751">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-812-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>u</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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>u</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></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><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>y</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-812">\begin{array}[t]{l}
 \forall u,a,e,t,attrs.~
 \\
 \qquad used(u,a,e,t,attrs)
 \\
 \quad\Rightarrow
 typeOf(a,activity) \wedge typeOf(e,entity)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-752-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>g</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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>g</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><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>y</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-752">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-813-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>g</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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>g</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><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>y</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-813">\begin{array}[t]{l}
 \forall g,a,e,t,attrs.~
 \\
 \qquad wasGeneratedBy(g,e,a,t,attrs)
 \\
 \quad\Rightarrow
 typeOf(a,activity) \wedge typeOf(e,entity)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-753-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>n</mi><mi>f</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>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>n</mi><mi>f</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>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><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><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-753">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-814-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>n</mi><mi>f</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>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>n</mi><mi>f</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>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><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><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-814">\begin{array}[t]{l}
 \forall inf,a_2,a_1,t,attrs.~
 \\
 \qquad wasInformedBy(inf,a_2,a_1,t,attrs)
 \\
 \quad\Rightarrow
 typeOf(a_1,activity) \wedge typeOf(a_2,activity)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-754-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><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><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><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><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-754">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-815-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><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><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><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><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-815">\begin{array}[t]{l}
 \forall start,a_2,e,a_1,t,attrs.~
 \\
 \qquad wasStartedBy(start,a_2,e,a_1,t,attrs)
 \\
 \quad\Rightarrow
 typeOf(a_1,activity) \wedge typeOf(a_2,activity) \wedge typeOf(e,entity)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-755-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>e</mi><mi>n</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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><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><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><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><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-755">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-816-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>e</mi><mi>n</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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>e</mi><mi>n</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><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><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>2</mn></msub><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><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-816">\begin{array}[t]{l}
 \forall end,a_2,e,a_1,t,attrs.~
 \\
 \qquad wasEndedBy(end,a_2,e,a_1,t,attrs)
 \\
 \quad\Rightarrow
 typeOf(a_1,activity) \wedge typeOf(a_2,activity) \wedge typeOf(e,entity)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-756-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>n</mi><mi>v</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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>n</mi><mi>v</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><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>y</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-756">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-817-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>n</mi><mi>v</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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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>n</mi><mi>v</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><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>y</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-817">\begin{array}[t]{l}
 \forall inv,a,e,t,attrs.~
 \\
 \qquad wasInvalidatedBy(inv,e,a,t,attrs)
 \\
 \quad\Rightarrow
 typeOf(a,activity) \wedge typeOf(e,entity)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-757-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>u</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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><msub><mi>u</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><mo stretchy="false">,</mo><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>u</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><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>y</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><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><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><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>y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-757">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-818-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>u</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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><msub><mi>u</mi><mn>1</mn></msub><mo stretchy="false">)</mo><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><mo stretchy="false">,</mo><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>u</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><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>y</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><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><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><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>y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-818">\begin{array}[t]{l}
 \forall id,e_2,e_1,a,g_2,u_1,attrs.~
 \\
 \qquad notNull(a) \wedge notNull(g_2) \wedge notNull(u_1) \wedge wasDerivedFrom(id,e_2,e_1,a,g_2,u_1,attrs)
 \\
 \quad\Rightarrow
 typeOf(e_2,entity) \wedge typeOf(e_1,activity) \wedge typeOf(a,activity)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-758-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>w</mi><mi>a</mi><mi>s</mi><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>r</mi><mi>o</mi><mi>m</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><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>y</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-758">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-819-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>w</mi><mi>a</mi><mi>s</mi><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>r</mi><mi>o</mi><mi>m</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mo stretchy="false">−</mo><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><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>y</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-819">\begin{array}[t]{l}
 \forall id,e_2,e_1,attrs.~
 \\
 \qquad wasDerivedFrom(id,e_2,e_1,-,-,-,attrs)
 \\
 \quad\Rightarrow
 typeOf(e_2,entity) \wedge typeOf(e_1,activity)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-759-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>T</mi><mi>o</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</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><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-759">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-820-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>i</mi><mi>b</mi><mi>u</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>T</mi><mi>o</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</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><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-820">\begin{array}[t]{l}
 \forall id,e,ag,attrs.~
 \\
 \qquad wasAttributedTo(id,e,ag,attrs)
 \\
 \quad\Rightarrow
 typeOf(e,entity) \wedge typeOf(ag,agent)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-760-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>p</mi><mi>l</mi><mo stretchy="false">)</mo><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></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><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>y</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>p</mi><mi>l</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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-760">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-821-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>p</mi><mi>l</mi><mo stretchy="false">)</mo><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></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><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>y</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>p</mi><mi>l</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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-821">\begin{array}[t]{l}
 \forall id,a,ag,pl,attrs.~
 \\
 \qquad notNull(pl) \wedge wasAssociatedWith(id,a,ag,pl,attrs)
 \\
 \quad\Rightarrow
 typeOf(a,activity) \wedge typeOf(ag,agent) \wedge typeOf(pl,entity)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-761-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><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>y</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-761">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-822-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><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>y</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-822">\begin{array}[t]{l}
 \forall id,a,ag,attrs.~
 \\
 \qquad wasAssociatedWith(id,a,ag,-,attrs)
 \\
 \quad\Rightarrow
 typeOf(a,activity) \wedge typeOf(ag,agent)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-762-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><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>y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-762">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-823-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><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><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><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><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><msub><mi>g</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><msub><mi>g</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><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>y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-823">\begin{array}[t]{l}
 \forall id,ag_2,ag_1,a,attrs.~
 \\
 \qquad actedOnBehalfOf(id,ag_2,ag_1,a,attrs)
 \\
 \quad\Rightarrow
 typeOf(ag_2,agent) \wedge typeOf(ag_1,agent) \wedge typeOf(a,activity)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-763-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><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>y</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-763">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-824-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><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>y</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-824">\begin{array}[t]{l}
 \forall e_2,e_1.~
 \\
 \qquad alternateOf(e_2,e_1)
 \\
 \quad\Rightarrow
 typeOf(e_2,entity) \wedge typeOf(e_1,entity)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-764-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><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>y</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-764">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-825-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><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>y</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-825">\begin{array}[t]{l}
 \forall e_2,e_1.~
 \\
 \qquad specializationOf(e_2,e_1)
 \\
 \quad\Rightarrow
 typeOf(e_2,entity) \wedge typeOf(e_1,entity)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-765-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>c</mi><mo stretchy="false">,</mo><mi>e</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>h</mi><mi>a</mi><mi>d</mi><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-765">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-826-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>c</mi><mo stretchy="false">,</mo><mi>e</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>h</mi><mi>a</mi><mi>d</mi><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</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></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-826">\begin{array}[t]{l}
 \forall c,e.~
 \\
 \qquad hadMember(c,e)
 \\
 \quad\Rightarrow
 typeOf(c,Collection) \wedge typeOf(e,entity)
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-766-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>c</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><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><mi>p</mi><mi>r</mi><mi>o</mi><mi>v</mi><mo stretchy="false">:</mo><mi>e</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>c</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><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>E</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-766">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-827-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>c</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>e</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>y</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><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><mi>p</mi><mi>r</mi><mi>o</mi><mi>v</mi><mo stretchy="false">:</mo><mi>e</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>c</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><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>E</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-827">\begin{array}[t]{l}
 \forall c.~
 \\
 \qquad entity(c,[prov:type = prov:emptyCollection]))
@@ -3500,21 +3577,21 @@
 <section id="impossibility-constraints">
 <h4><span class="secno">5.2.4 </span>Impossibility constraints</h4>
 
-<div class="constraint" id="impossible-unspecified-derivation-generation-use"><div class="ruleTitle"><a class="internalDFN" href="#impossible-unspecified-derivation-generation-use">Constraint 51 (impossible-unspecified-derivation-generation-use)</a></div><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-767-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><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><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></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-767">\begin{array}[t]{l}
+<div class="constraint" id="impossible-unspecified-derivation-generation-use"><div class="ruleTitle"><a class="internalDFN" href="#impossible-unspecified-derivation-generation-use">Constraint 51 (impossible-unspecified-derivation-generation-use)</a></div><ol><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-828-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><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><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></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-828">\begin{array}[t]{l}
 \forall id,e_1,e_2,g,attrs.~
 \\
 \qquad notNull(g) \wedge wasDerivedFrom(id,e_2,e_1,-,g,-,attrs)
 \\
 \quad\Rightarrow
 False
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-768-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>u</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><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><mi>u</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-768">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-829-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>u</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><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><mi>u</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-829">\begin{array}[t]{l}
 \forall id,e_1,e_2,u,attrs.~
 \\
 \qquad notNull(u) \wedge wasDerivedFrom(id,e_2,e_1,-,-,u,attrs)
 \\
 \quad\Rightarrow
 False
-\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-769-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><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><mi>g</mi><mo stretchy="false">,</mo><mi>u</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-769">\begin{array}[t]{l}
+\end{array}</script></li><li><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-830-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">,</mo><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>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>n</mi><mi>o</mi><mi>t</mi><mi>N</mi><mi>u</mi><mi>l</mi><mi>l</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo><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><mi>g</mi><mo stretchy="false">,</mo><mi>u</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-830">\begin{array}[t]{l}
 \forall id,e_1,e_2,g,u,attrs.~
 \\
 \qquad notNull(g) \wedge notNull(u) \wedge wasDerivedFrom(id,e_2,e_1,-,g,u,attrs)
@@ -3524,10 +3601,10 @@
 \end{array}</script></li></ol></div>
 <div class="proof">
   <p>Each part follows from the fact that the semantics of
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-770-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mi>a</mi><mi>s</mi><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>r</mi><mi>o</mi><mi>m</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-770">wasDerivedFrom</script> only allows formulas to hold when either all three
-  of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-771-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo stretchy="false">,</mo><mi>g</mi><mo stretchy="false">,</mo><mi>u</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-771">a,g,u</script> are <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-772-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">−</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-772">-</script> (denoting <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-773-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">⊥</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-773">\bot</script>) or none of them are.</p>
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-831-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mi>a</mi><mi>s</mi><mi>D</mi><mi>e</mi><mi>r</mi><mi>i</mi><mi>v</mi><mi>e</mi><mi>d</mi><mi>F</mi><mi>r</mi><mi>o</mi><mi>m</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-831">wasDerivedFrom</script> only allows formulas to hold when either all three
+  of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-832-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo stretchy="false">,</mo><mi>g</mi><mo stretchy="false">,</mo><mi>u</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-832">a,g,u</script> are <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-833-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">−</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-833">-</script> (denoting <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-834-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">⊥</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-834">\bot</script>) or none of them are.</p>
   </div>
-<div class="constraint" id="impossible-specialization-reflexive"><div class="ruleTitle"><a class="internalDFN" href="#impossible-specialization-reflexive">Constraint 52 (impossible-specialization-reflexive)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-774-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>e</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-774">\begin{array}[t]{l}
+<div class="constraint" id="impossible-specialization-reflexive"><div class="ruleTitle"><a class="internalDFN" href="#impossible-specialization-reflexive">Constraint 52 (impossible-specialization-reflexive)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-835-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>e</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-835">\begin{array}[t]{l}
 \forall e.~
 \\
 \qquad specializationOf(e,e)
@@ -3537,11 +3614,11 @@
 \end{array}</script></div>
 <div class="proof">
   <p>This follows from the fact that in the semantics of
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-775-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-775">specializationOf</script>, the two entities denoted by the first and second
-  arguments are required to be distinct.
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-836-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-836">specializationOf</script>, the requirement that one of the inclusions is
+  strict implies that the two entities cannot be the same.
   </p>
   </div>
-<div class="constraint" id="impossible-property-overlap"><div class="ruleTitle"><a class="internalDFN" href="#impossible-property-overlap">Constraint 53 (impossible-property-overlap)</a></div><p>For each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-776-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-776">r</script>  and  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-777-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">∈</mo><mo fence="false" stretchy="false">{</mo><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</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>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>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>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>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>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>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>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 fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-777">s \in \{ used, wasGeneratedBy, wasInvalidatedBy, wasStartedBy, wasEndedBy, wasInformedBy, wasAttributedTo, wasAssociatedWith, actedOnBehalfOf\}</script> such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-778-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-778">r</script>  and  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-779-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-779">s</script> are different relation names, the following constraint holds:</p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-780-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>m</mi></msub><mo stretchy="false">,</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>b</mi><mi>n</mi></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>m</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>s</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-780">\begin{array}[t]{l}
+<div class="constraint" id="impossible-property-overlap"><div class="ruleTitle"><a class="internalDFN" href="#impossible-property-overlap">Constraint 53 (impossible-property-overlap)</a></div><p>For each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-837-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-837">r</script>  and  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-838-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">∈</mo><mo fence="false" stretchy="false">{</mo><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</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>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>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>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>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>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>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>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 fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-838">s \in \{ used, wasGeneratedBy, wasInvalidatedBy, wasStartedBy, wasEndedBy, wasInformedBy, wasAttributedTo, wasAssociatedWith, actedOnBehalfOf\}</script> such that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-839-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-839">r</script>  and  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-840-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-840">s</script> are different relation names, the following constraint holds:</p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-841-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>m</mi></msub><mo stretchy="false">,</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>b</mi><mi>n</mi></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>m</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>s</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-841">\begin{array}[t]{l}
 \forall id,a_1,\ldots,a_m,b_1,\ldots,b_n.~
 \\
 \qquad r(id,a_1,\ldots,a_m) \wedge s(id,b_1,\ldots,b_n)
@@ -3550,11 +3627,11 @@
 False
 \end{array}</script></div>
 <div class="proof">
-  <p>This follows from the assumption that the different classes of
+  <p>This follows from the assumption that the different kinds of
   influences are disjoint sets, characterized by their types.
   </p>
   </div>
-<div class="constraint" id="impossible-object-property-overlap"><div class="ruleTitle"><a class="internalDFN" href="#impossible-object-property-overlap">Constraint 54 (impossible-object-property-overlap)</a></div><p>For each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-781-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo stretchy="false">∈</mo><mo fence="false" 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>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>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-781">p \in \{entity,activity,agent\}</script>  and each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-782-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo stretchy="false">∈</mo><mo fence="false" stretchy="false">{</mo><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</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>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>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>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>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>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>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>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 fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-782">r \in \{ used, wasGeneratedBy, wasInvalidatedBy, wasStartedBy, wasEndedBy, wasInformedBy, wasAttributedTo, wasAssociatedWith, actedOnBehalfOf\}</script>, the following constraint holds:</p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-783-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>m</mi></msub><mo stretchy="false">,</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>b</mi><mi>n</mi></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>p</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>m</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-783">\begin{array}[t]{l}
+<div class="constraint" id="impossible-object-property-overlap"><div class="ruleTitle"><a class="internalDFN" href="#impossible-object-property-overlap">Constraint 54 (impossible-object-property-overlap)</a></div><p>For each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-842-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo stretchy="false">∈</mo><mo fence="false" 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>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>a</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-842">p \in \{entity,activity,agent\}</script>  and each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-843-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo stretchy="false">∈</mo><mo fence="false" stretchy="false">{</mo><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</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>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>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>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>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>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>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>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 fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-843">r \in \{ used, wasGeneratedBy, wasInvalidatedBy, wasStartedBy, wasEndedBy, wasInformedBy, wasAttributedTo, wasAssociatedWith, actedOnBehalfOf\}</script>, the following constraint holds:</p><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-844-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>m</mi></msub><mo stretchy="false">,</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>b</mi><mi>n</mi></msub><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>p</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>m</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-844">\begin{array}[t]{l}
 \forall id,a_1,\ldots,a_m,b_1,\ldots,b_n.~
 \\
 \qquad p(id,a_1,\ldots,a_m) \wedge r(id,b_1,\ldots,b_n)
@@ -3567,7 +3644,7 @@
   from other objects (entities, activities or agents).
   </p>
   </div>
-<div class="constraint" id="entity-activity-disjoint"><div class="ruleTitle"><a class="internalDFN" href="#entity-activity-disjoint">Constraint 55 (entity-activity-disjoint)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-784-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</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><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</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></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-784">\begin{array}[t]{l}
+<div class="constraint" id="entity-activity-disjoint"><div class="ruleTitle"><a class="internalDFN" href="#entity-activity-disjoint">Constraint 55 (entity-activity-disjoint)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-845-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>i</mi><mi>d</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</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><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</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></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-845">\begin{array}[t]{l}
 \forall id.~
 \\
 \qquad typeOf(id,entity) \wedge typeOf(id,activity)
@@ -3579,7 +3656,7 @@
   <p>This follows from the assumption that entities and activities are disjoint.
   </p>
   </div>
-<div class="constraint" id="membership-empty-collection"><div class="ruleTitle"><a class="internalDFN" href="#membership-empty-collection">Constraint 56 (membership-empty-collection)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-785-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>c</mi><mo stretchy="false">,</mo><mi>e</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>h</mi><mi>a</mi><mi>s</mi><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>E</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-785">\begin{array}[t]{l}
+<div class="constraint" id="membership-empty-collection"><div class="ruleTitle"><a class="internalDFN" href="#membership-empty-collection">Constraint 56 (membership-empty-collection)</a></div><span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-846-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable align="baseline 1" columnalign="left" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mi mathvariant="normal">∀</mi><mi>c</mi><mo stretchy="false">,</mo><mi>e</mi><mo>.</mo><mtext>&nbsp;</mtext></mtd></mtr><mtr><mtd><mspace width="2em"></mspace><mi>h</mi><mi>a</mi><mi>s</mi><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>t</mi><mi>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>E</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mspace width="1em"></mspace><mo stretchy="false">⇒</mo><mi>F</mi><mi>a</mi><mi>l</mi><mi>s</mi><mi>e</mi></mtd></mtr></mtable></math></span></span></span><script type="math/tex" id="MathJax-Element-846">\begin{array}[t]{l}
 \forall c,e.~
 \\
 \qquad hasMember(c,e) \wedge typeOf(c,EmptyCollection)
@@ -3589,8 +3666,8 @@
 \end{array}</script></div>
 <div class="proof">
   <p>This follows from the definition of the semantics of
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-786-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>E</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-786">typeOf(c,EmptyCollection)</script>, which requires that there are no
-  members of the collection denoted by <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-787-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-787">c</script>.
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-847-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>y</mi><mi>p</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">,</mo><mi>E</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-847">typeOf(c,EmptyCollection)</script>, which requires that there are no
+  members of the collection denoted by <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-848-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-848">c</script>.
   </p>
   </div>
 </section>
@@ -3613,33 +3690,35 @@
 <p>Our main soundness result is:</p>
 
 <div class="theorem" id="soundness-theorem" data-count="39" data-title="Theorem 39 (soundness-theorem)"><div class="ruleTitle"><a class="internalDFN" href="#soundness-theorem">Theorem 39 (soundness-theorem)</a></div>
-  <p> Let <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-788-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-788">W</script> be a PROV model, that is, a structure satisfying all of
-  the axioms</p>.
+  <p> Let <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-849-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-849">W</script> be a PROV structure, that is, a structure providing all
+  of the components above and satisfying all of
+  the axioms.</p>
   <ol>
-  <li>If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-789-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-789">I</script> is an instance and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-790-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-790">W \models I</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-791-Frame" style="font-size: 100%; "><span class="MathJax_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>I</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-791">I'</script> is obtained from <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-792-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-792">I</script> by applying one
-  of the PROV inferences, then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-793-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msup><mi>I</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-793">W \models I'</script>.</li>
-  <li>If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-794-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-794">I</script> is an instance and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-795-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-795">W \models I</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-796-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-796">I</script> has a normal
-  form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-797-Frame" style="font-size: 100%; "><span class="MathJax_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>I</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-797">I'</script> and   <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-798-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msup><mi>I</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-798"> W \models I'</script>.</li>
-  <li>If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-799-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-799">I</script> is a normal form and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-800-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-800">W \models I</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-801-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-801">I</script> satisfies all of the ordering,
+  <li>If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-850-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-850">I</script> is an instance and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-851-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-851">W \models I</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-852-Frame" style="font-size: 100%; "><span class="MathJax_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>I</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-852">I'</script> is obtained from <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-853-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-853">I</script> by applying one
+  of the PROV inferences, then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-854-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msup><mi>I</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-854">W \models I'</script>.</li>
+  <li>If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-855-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-855">I</script> is an instance and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-856-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-856">W \models I</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-857-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-857">I</script> has a normal
+  form <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-858-Frame" style="font-size: 100%; "><span class="MathJax_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>I</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-858">I'</script> and   <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-859-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msup><mi>I</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-859"> W \models I'</script>.</li>
+  <li>If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-860-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-860">I</script> is a normal form and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-861-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-861">W \models I</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-862-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-862">I</script> satisfies all of the ordering,
   typing and impossibility constraints.
-  </li><li>If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-802-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-802">W \models I</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-803-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-803">I</script> is valid.</li>
+  </li><li>If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-863-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-863">W \models I</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-864-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-864">I</script> is valid.</li>
 </ol>
   </div>
 <div class="proof">
   <p>For part 1, the arguments are as in the previous section.  </p>
   <p>For
   part 2, proceed by induction on a terminating sequence of inference
-  or uniqueness constraint steps: if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-804-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-804">I</script> is in normal form then we are
-  done. If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-805-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-805">I</script> is not in normal form then if an inference is  applicable, then use part 1; if a uniqueness constraint is
-  applicable, then from <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-806-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-806">W \models I</script> the uniqueness constraint cannot
-  fail on <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-807-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-807">I</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-808-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msup><mi>I</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-808">W \models I'</script>.</p>
+  or uniqueness constraint steps: if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-865-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-865">I</script> is in normal form then we are
+  done. If <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-866-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-866">I</script> is not in normal form then if an inference is  applicable, then use part 1; if a uniqueness constraint is
+  applicable, then from <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-867-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-867">W \models I</script> the uniqueness constraint cannot
+  fail on <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-868-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-868">I</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-869-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msup><mi>I</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-869">W \models I'</script>, as argued for the key constraints in
+  the previous section.</p>
 <p>
   For part 3, the arguments are as
   in the previous section for each constraint. </p>
   <p>Finally, for part 4,
-  suppose <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-809-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-809">W \models I</script>.  Then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-810-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msup><mi>I</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-810">W \models I'</script> where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-811-Frame" style="font-size: 100%; "><span class="MathJax_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>I</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-811">I'</script> is the normal
-  form of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-812-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-812">I</script> by part 2.  By part 3, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-813-Frame" style="font-size: 100%; "><span class="MathJax_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>I</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-813">I'</script> satisfies all of the
-  remaining constraints, so <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-814-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-814">I</script> is valid.</p>
+  suppose <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-870-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-870">W \models I</script>.  Then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-871-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><msup><mi>I</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-871">W \models I'</script> where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-872-Frame" style="font-size: 100%; "><span class="MathJax_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>I</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-872">I'</script> is the normal
+  form of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-873-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-873">I</script> by part 2.  By part 3, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-874-Frame" style="font-size: 100%; "><span class="MathJax_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>I</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-874">I'</script> satisfies all of the
+  remaining constraints, so <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-875-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-875">I</script> is valid.</p>
   </div>
 
   </section>
@@ -3650,15 +3729,15 @@
   structures, and show that a valid PROV instance has a model.</p>
 
   
-  <p> Let <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-815-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-815">I</script> be a valid PROV instance that is in normal form.
-  We define a structure <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-816-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-816">M(I)</script> as follows, by giving the sets,
-  functions and relations specified in the components in <a href="#structures">Section 3</a>.</p>
+  <p> Let <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-876-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-876">I</script> be a valid PROV instance that is in normal form.
+  We define a structure <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-877-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-877">M(I)</script> as follows, by giving the sets,
+  functions and relations specified in the components in <a href="#structures">Section 3</a>, and finally verifying that the
+  axioms hold.</p>
 
   <section id="sets">
 <h4><span class="secno">6.2.1 </span>Sets</h4>
-  <p> The sets of structure <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-817-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-817">M(I)</script> are: </p>
-
-  <span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-818-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="thickmathspace" displaystyle="true" side="right" minlabelspacing="0.8em"><mtr><mtd><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>P</mi><mi>l</mi><mi>a</mi><mi>n</mi><mi>s</mi></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>p</mi><mi>l</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><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">≠</mo><mo stretchy="false">−</mo><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>e</mi><mo stretchy="false">∣</mo><mi>m</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">∪</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>e</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>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</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><mi>p</mi><mi>r</mi><mi>o</mi><mi>v</mi><mo stretchy="false">:</mo><mi>e</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">∈</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">∪</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><msub><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>d</mi></mrow></msub><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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>A</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mi>U</mi><mi>s</mi><mi>a</mi><mi>g</mi><mi>e</mi><mi>s</mi></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">∪</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>d</mi></mrow></msub><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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">∪</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><msub><mi>g</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>d</mi></mrow></msub><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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>S</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>s</mi></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><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><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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>E</mi><mi>n</mi><mi>d</mi><mi>s</mi></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><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><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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mi>U</mi><mi>s</mi><mi>a</mi><mi>g</mi><mi>e</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>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>S</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>E</mi><mi>n</mi><mi>d</mi><mi>s</mi></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>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>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>D</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">∪</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</mi><mo stretchy="false">∣</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>q</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>q</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>A</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">∪</mo><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></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-818">
+  <p> The sets of structure <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-878-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-878">M(I)</script> are: </p>
+<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-879-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="thickmathspace" displaystyle="true" side="right" minlabelspacing="0.8em"><mtr><mtd><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>P</mi><mi>l</mi><mi>a</mi><mi>n</mi><mi>s</mi></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>p</mi><mi>l</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><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">≠</mo><mo stretchy="false">−</mo><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>e</mi><mo stretchy="false">∣</mo><mi>m</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">∪</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>e</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>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</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><mi>p</mi><mi>r</mi><mi>o</mi><mi>v</mi><mo stretchy="false">:</mo><mi>e</mi><mi>m</mi><mi>p</mi><mi>t</mi><mi>y</mi><mi>C</mi><mi>o</mi><mi>l</mi><mi>l</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mo stretchy="false">∈</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">∪</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><msub><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>d</mi></mrow></msub><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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>A</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mi>U</mi><mi>s</mi><mi>a</mi><mi>g</mi><mi>e</mi><mi>s</mi></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">∪</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>d</mi></mrow></msub><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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">∪</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><msub><mi>g</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>d</mi></mrow></msub><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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>S</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>s</mi></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><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><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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>E</mi><mi>n</mi><mi>d</mi><mi>s</mi></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><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><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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mi>U</mi><mi>s</mi><mi>a</mi><mi>g</mi><mi>e</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>G</mi><mi>e</mi><mi>n</mi><mi>e</mi><mi>r</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>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>S</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>E</mi><mi>n</mi><mi>d</mi><mi>s</mi></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>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>C</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>D</mi><mi>e</mi><mi>l</mi><mi>e</mi><mi>g</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">∪</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</mi><mo stretchy="false">∣</mo><mi>w</mi><mi>a</mi><mi>s</mi><mi>I</mi><mi>n</mi><mi>f</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>n</mi><mi>c</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>q</mi><mn>2</mn></msub><mo stretchy="false">,</mo><msub><mi>q</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><mi>O</mi><mi>b</mi><mi>j</mi><mi>e</mi><mi>c</mi><mi>t</mi><mi>s</mi></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mi>E</mi><mi>n</mi><mi>t</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi><mo stretchy="false">∪</mo><mi>A</mi><mi>g</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">∪</mo><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></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-879">
   \begin{eqnarray*}
   Entities &=& \{id \mid entity(id,attrs) \in I\}\\
   Plans &=& \{pl \mid wasAssociatedWith(id,ag,act,pl,attrs) \in I, pl
@@ -3693,28 +3772,28 @@
   Objects &=& Entities \cup Activities \cup Agents \cup Influences\\
   \end{eqnarray*}
   </script>
-<p>In the definitions of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-819-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-819">Activities</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-820-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-820">Generations</script>, and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-821-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>a</mi><mi>g</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-821">Usages</script> we
-  write <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-822-Frame" style="font-size: 100%; "><span class="MathJax_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><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>d</mi></mrow></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-822">a_{id}</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-823-Frame" style="font-size: 100%; "><span class="MathJax_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><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>d</mi></mrow></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-823">g_{id}</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-824-Frame" style="font-size: 100%; "><span class="MathJax_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><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>d</mi></mrow></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-824">u_{id}</script> respectively to indicate
+<p>In the definitions of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-880-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>c</mi><mi>t</mi><mi>i</mi><mi>v</mi><mi>i</mi><mi>t</mi><mi>i</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-880">Activities</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-881-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>i</mi><mi>o</mi><mi>n</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-881">Generations</script>, and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-882-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>a</mi><mi>g</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-882">Usages</script> we
+  write <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-883-Frame" style="font-size: 100%; "><span class="MathJax_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><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>d</mi></mrow></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-883">a_{id}</script>, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-884-Frame" style="font-size: 100%; "><span class="MathJax_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><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>d</mi></mrow></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-884">g_{id}</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-885-Frame" style="font-size: 100%; "><span class="MathJax_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><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>d</mi></mrow></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-885">u_{id}</script> respectively to indicate
   additional activities, generations and usages added for imprecise derivations.
 </p>
-  <p> In addition, to define the set of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-825-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-825">Things</script>, we introduce an
-  equivalence relation on <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-826-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-826">Entities</script> as follows:
+  <p> In addition, to define the set of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-886-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-886">Things</script>, we introduce an
+  equivalence relation on <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-887-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-887">Entities</script> as follows:
   </p>
-  <span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-827-Frame" style="font-size: 100%; "><span class="MathJax_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"><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">≡</mo><msub><mi>e</mi><mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo stretchy="false">⟺</mo><mspace width="thickmathspace"></mspace><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-827">e_1 \equiv e_2 \iff alternateOf(e_1,e_2) \in I</script>
+  <span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-888-Frame" style="font-size: 100%; "><span class="MathJax_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"><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">≡</mo><msub><mi>e</mi><mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo stretchy="false">⟺</mo><mspace width="thickmathspace"></mspace><mi>a</mi><mi>l</mi><mi>t</mi><mi>e</mi><mi>r</mi><mi>n</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msub><mi>e</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-888">e_1 \equiv e_2 \iff alternateOf(e_1,e_2) \in I</script>
   <p>The fact that this is an equivalence relation follows from the
-  fact that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-828-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-828">I</script> is in normal form, since the constraints on
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-829-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-829">alternateOf</script> ensure that it is an equivalence relation.  Recall
-  that given an equivalence relation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-830-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">≡</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-830">\equiv</script> on some set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-831-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-831">X</script>, the
-  <em>equivalence class</em> of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-832-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">∈</mo><mi>X</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-832">x \in X</script> is the set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-833-Frame" style="font-size: 100%; "><span class="MathJax_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>x</mi><msub><mo stretchy="false">]</mo><mo stretchy="false">≡</mo></msub><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><mi>y</mi><mo stretchy="false">∈</mo><mi>X</mi><mo stretchy="false">∣</mo><mi>x</mi><mo stretchy="false">≡</mo><mi>y</mi><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-833">[x]_\equiv = \{y
+  fact that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-889-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-889">I</script> is in normal form, since the constraints on
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-890-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-890">alternateOf</script> ensure that it is an equivalence relation.  Recall
+  that given an equivalence relation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-891-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">≡</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-891">\equiv</script> on some set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-892-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-892">X</script>, the
+  <em>equivalence class</em> of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-893-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">∈</mo><mi>X</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-893">x \in X</script> is the set <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-894-Frame" style="font-size: 100%; "><span class="MathJax_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>x</mi><msub><mo stretchy="false">]</mo><mo stretchy="false">≡</mo></msub><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><mi>y</mi><mo stretchy="false">∈</mo><mi>X</mi><mo stretchy="false">∣</mo><mi>x</mi><mo stretchy="false">≡</mo><mi>y</mi><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-894">[x]_\equiv = \{y
   \in X \mid x
-  \equiv y\}</script>.  The <em>quotient</em> of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-834-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-834">X</script> by an equivalence
-  relation on <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-835-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-835">X</script> is the set of equivalence classes, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-836-Frame" style="font-size: 100%; "><span class="MathJax_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><mo stretchy="false">≡</mo></msub><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><mo stretchy="false">[</mo><mi>x</mi><msub><mo stretchy="false">]</mo><mo stretchy="false">≡</mo></msub><mo stretchy="false">∣</mo><mi>x</mi><mo stretchy="false">∈</mo><mi>X</mi><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-836">X_\equiv =
+  \equiv y\}</script>.  The <em>quotient</em> of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-895-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-895">X</script> by an equivalence
+  relation on <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-896-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-896">X</script> is the set of equivalence classes, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-897-Frame" style="font-size: 100%; "><span class="MathJax_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><mo stretchy="false">≡</mo></msub><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><mo stretchy="false">[</mo><mi>x</mi><msub><mo stretchy="false">]</mo><mo stretchy="false">≡</mo></msub><mo stretchy="false">∣</mo><mi>x</mi><mo stretchy="false">∈</mo><mi>X</mi><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-897">X_\equiv =
   \{[x]_\equiv \mid x \in X\}</script>.  Now we
-  define the set of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-837-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-837">Things</script> as the quotient of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-838-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">≡</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-838">\equiv</script>-equivalence classes of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-839-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-839">Entities</script>.</p>
-
-<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-840-Frame" style="font-size: 100%; "><span class="MathJax_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>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</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><msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">≡</mo></mrow></msub><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><mo stretchy="false">[</mo><mi>e</mi><msub><mo stretchy="false">]</mo><mo stretchy="false">≡</mo></msub><mo stretchy="false">∣</mo><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><mo fence="false" stretchy="false">}</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-840">Things = Entities /_{\equiv} = \{[e]_\equiv \mid e \in Entities\}</script>
-
-<p> Observe that since <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-841-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-841">I</script> is normalized and valid, entities and
+  define the set of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-898-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-898">Things</script> as the quotient of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-899-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">≡</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-899">\equiv</script>-equivalence classes of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-900-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-900">Entities</script>.</p>
+
+<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-901-Frame" style="font-size: 100%; "><span class="MathJax_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>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</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><msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">≡</mo></mrow></msub><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><mo stretchy="false">[</mo><mi>e</mi><msub><mo stretchy="false">]</mo><mo stretchy="false">≡</mo></msub><mo stretchy="false">∣</mo><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><mo fence="false" stretchy="false">}</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-901">Things = Entities /_{\equiv} = \{[e]_\equiv \mid e \in Entities\}</script>
+
+<p> Observe that since <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-902-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-902">I</script> is normalized and valid, entities and
 activities are disjoint, the influences are disjoint from entities,
 activities, and agents, 
 and the different subsets of events and influences are pairwise
@@ -3723,25 +3802,39 @@
 <section id="functions">
 <h4><span class="secno">6.2.2 </span>Functions</h4>
 
-<p> First, we consider the functions associated with <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-842-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-842">Entities</script>.  We
-employ an auxiliary function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-843-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">:</mo><mi>E</mi><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>P</mi><mo stretchy="false">(</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-843">events:Entities \to P(Events)</script> that collects the set of all
-events in which <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-844-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-844">e</script> participated.</p>
-<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-845-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="thickmathspace" displaystyle="true" side="right" minlabelspacing="0.8em"><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">∪</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">∪</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">∪</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><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><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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">∪</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><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><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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">∣</mo><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><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>v</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>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">=</mo><mi>v</mi><mo stretchy="false">∈</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo stretchy="false">[</mo><mi>e</mi><msub><mo stretchy="false">]</mo><mo stretchy="false">≡</mo></msub></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-845">
+<p> First, we consider the functions associated with <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-903-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-903">Entities</script>.  We
+employ an auxiliary function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-904-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">:</mo><mi>E</mi><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>P</mi><mo stretchy="false">(</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-904">events:Entities \to P(Events)</script> that collects the set of all
+events in which <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-905-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-905">e</script> participated.</p>
+<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-906-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="thickmathspace" displaystyle="true" side="right" minlabelspacing="0.8em"><mtr><mtd><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><msup><mi>s</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">∪</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">∪</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">∪</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><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><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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo stretchy="false">∪</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>i</mi><mi>d</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><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><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><mo stretchy="false">∈</mo><mi>I</mi><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><msup><mi>s</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">∪</mo><munder><mo stretchy="false">⋃</mo><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi></mrow></munder><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><msup><mi>s</mi><mo>′</mo></msup><mo stretchy="false">(</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">∣</mo><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><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>v</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>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi><mo stretchy="false">,</mo><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">=</mo><mi>v</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo fence="false" stretchy="false">}</mo><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">≠</mo><mi>u</mi><mi>n</mi><mi>i</mi><mi>q</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>u</mi><mi>n</mi><mi>i</mi><mi>q</mi><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>u</mi><mi>n</mi><mi>i</mi><msub><mi>q</mi><mrow class="MJX-TeXAtom-ORD"><mi>e</mi></mrow></msub><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">∪</mo><munder><mo stretchy="false">⋃</mo><mrow class="MJX-TeXAtom-ORD"><mi>s</mi><mi>p</mi><mi>e</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>l</mi><mi>i</mi><mi>z</mi><mi>a</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi></mrow></munder><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">(</mo><msup><mi>e</mi><mo>′</mo></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mi>t</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo stretchy="false">[</mo><mi>e</mi><msub><mo stretchy="false">]</mo><mo stretchy="false">≡</mo></msub></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-906">
 \begin{eqnarray*}
-events(e) &=& \{id \mid used(id,a,e,t,attrs) \in I\}\\
+events'(e) &=& \{id \mid used(id,a,e,t,attrs) \in I\}\\
 &\cup& \{id \mid wasGeneratedBy(id,e,a,t,attrs) \in I\}\\
 &\cup& \{id \mid wasInvalidatedBy(id,e,a,t,attrs) \in I\}\\
 &\cup& \{id \mid wasStartedBy(id,a,e,a',t,attrs) \in I\}\\
 &\cup& \{id \mid wasEndedBy(id,a,e,a',t,attrs) \in I\}\\
+events(e) &=& events'(e) \cup \bigcup_{specializationOf(e',e) \in I} events'(e')\\
 lifetime(e) &=& \{time(e) \mid evt \in events(e)\}\\
-value(e,a) &=& \{v \mid entity(e,attrs) \in I, a=v \in attrs\}\\
+value'(e,a) &=& \{v \mid entity(e,attrs) \in I, (a=v) \in attrs\}
+\quad (a \neq uniq)\\
+value'(e,uniq) &=&\{ uniq_{e}\}\\
+value(e,a) &=&  value'(e) \cup \bigcup_{specializationOf(e',e) \in I} value'(e')\\
 thingOf(e) &=& [e]_\equiv
 \end{eqnarray*}
 </script>
-<p> Similarly, for <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-846-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-846">Things</script>, we
-employ an auxiliary function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-847-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">:</mo><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi><mo stretchy="false">→</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-847">events:Things \to P(Events)</script> that collects the set of all
+<p>Above, we introduce a fresh attribute name <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-907-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>n</mi><mi>i</mi><mi>q</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-907">uniq</script>, not already in
+use in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-908-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-908">I</script>, along with a fresh value <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-909-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-909">e</script> and for each entity <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-910-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-910">e</script> we
+add an attribute-value pair <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-911-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>n</mi><mi>i</mi><mi>q</mi><mo stretchy="false">=</mo><mi>u</mi><mi>n</mi><mi>i</mi><msub><mi>q</mi><mi>e</mi></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-911">uniq=uniq_e</script> to <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-912-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mi>s</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>u</mi><mi>n</mi><mi>i</mi><mi>q</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-912">values(e,uniq)</script>.  This
+construction ensures that if an entity is a specialization of another
+in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-913-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-913">I</script> then the specialization relationship will hold in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-914-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-914">M(I)</script>.  We
+also define the set of all events involved in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-915-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-915">e</script> as the set of events
+immediately involved in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-916-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-916">e</script> or any specialization of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-917-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-917">e</script>.  Similarly,
+the values of attributes of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-918-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-918">e</script> are those immediately declared for <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-919-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-919">e</script>
+along with those of any specialization.
+</p>
+<p> Similarly, for <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-920-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-920">Things</script>, we
+employ an auxiliary function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-921-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">:</mo><mi>T</mi><mi>h</mi><mi>i</mi><mi>n</mi><mi>g</mi><mi>s</mi><mo stretchy="false">→</mo><mi>P</mi><mo stretchy="false">(</mo><mi>E</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-921">events:Things \to P(Events)</script> that collects the set of all
 events in which one of the entities constituting the thing participated.</p>
-<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-848-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="thickmathspace" displaystyle="true" side="right" minlabelspacing="0.8em"><mtr><mtd><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><munder><mo stretchy="false">⋃</mo><mrow class="MJX-TeXAtom-ORD"><mi>e</mi><mo stretchy="false">∈</mo><mi>T</mi></mrow></munder><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></mtd></mtr><mtr><mtd><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">∣</mo><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><munder><mo stretchy="false">⋃</mo><mrow class="MJX-TeXAtom-ORD"><mi>e</mi><mo stretchy="false">∈</mo><mi>T</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">∈</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></munder><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></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-848">
+<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-922-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="thickmathspace" displaystyle="true" side="right" minlabelspacing="0.8em"><mtr><mtd><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><munder><mo stretchy="false">⋃</mo><mrow class="MJX-TeXAtom-ORD"><mi>e</mi><mo stretchy="false">∈</mo><mi>T</mi></mrow></munder><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></mtd></mtr><mtr><mtd><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo fence="false" stretchy="false">{</mo><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">∣</mo><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">∈</mo><mi>e</mi><mi>v</mi><mi>e</mi><mi>n</mi><mi>t</mi><mi>s</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo></mtd></mtr><mtr><mtd><mi>v</mi><mi>a</mi><mi>l</mi><mi>u</mi><mi>e</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><munder><mo stretchy="false">⋃</mo><mrow class="MJX-TeXAtom-ORD"><mi>e</mi><mo stretchy="false">∈</mo><mi>T</mi><mo stretchy="false">,</mo><mi>t</mi><mo stretchy="false">∈</mo><mi>l</mi><mi>i</mi><mi>f</mi><mi>e</mi><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></munder><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></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-922">
 \begin{eqnarray*}
 events(T) &=& \bigcup_{e \in T} events(e)\\
 lifetime(e) &=& \{time(e) \mid evt \in events(T)\}\\
@@ -3754,24 +3847,25 @@
   some of the times are symbolic?</p>
   </div></div>
 
-<p> The functions <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-849-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-849">startTime</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-850-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-850">endTime</script> mapping activities to
+<p> The functions <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-923-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-923">startTime</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-924-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-924">endTime</script> mapping activities to
   their start and end times is defined as follows:
   </p>
-  <span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-851-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><span style="display: inline-block; vertical-align: middle; font-size: 90%; text-align: left; color: black; padding: 1px 3px; border: 1px solid; ">\begin{eqnarray*}<br>&nbsp;&nbsp;startTime(id)&nbsp;&amp;=&amp;&nbsp;t_1&nbsp;\text{&nbsp;where&nbsp;}&nbsp;activity(a,t_1,t_2,attrs)&nbsp;\in<br>&nbsp;&nbsp;I\\<br>&nbsp;&nbsp;&amp;&nbsp;\text{&nbsp;or&nbsp;}&nbsp;wasStartedBy(start,a,e,a',t_1,attrs)&nbsp;\in&nbsp;I\\<br>&nbsp;&nbsp;endTime(id)&nbsp;&amp;=&amp;&nbsp;t_2&nbsp;\text{&nbsp;where&nbsp;}&nbsp;activity(a,t_1,t_2,attrs)&nbsp;\in<br>&nbsp;&nbsp;I\\<br>&nbsp;&nbsp;&amp;&nbsp;\text{&nbsp;or&nbsp;}&nbsp;wasEndedBy(end,a,e,a',t_2,attrs)&nbsp;\in&nbsp;I\\</span></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-851">\begin{eqnarray*}
+  <span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-925-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="thickmathspace" displaystyle="true" side="right" minlabelspacing="0.8em"><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><msub><mi>t</mi><mn>1</mn></msub><mtext>&nbsp;where&nbsp;</mtext><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>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd></mtd><mtd><mtext>&nbsp;or&nbsp;</mtext><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>s</mi><mi>t</mi><mi>a</mi><mi>r</mi><mi>t</mi><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><msub><mi>t</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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><msub><mi>t</mi><mn>2</mn></msub><mtext>&nbsp;where&nbsp;</mtext><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>a</mi><mo stretchy="false">,</mo><msub><mi>t</mi><mn>1</mn></msub><mo stretchy="false">,</mo><msub><mi>t</mi><mn>2</mn></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd></mtd><mtd><mtext>&nbsp;or&nbsp;</mtext><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>e</mi><mi>n</mi><mi>d</mi><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><msub><mi>t</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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-925">\begin{eqnarray*}
   startTime(id) &=& t_1 \text{ where } activity(a,t_1,t_2,attrs) \in
   I\\
   & \text{ or } wasStartedBy(start,a,e,a',t_1,attrs) \in I\\
   endTime(id) &=& t_2 \text{ where } activity(a,t_1,t_2,attrs) \in
   I\\
   & \text{ or } wasEndedBy(end,a,e,a',t_2,attrs) \in I\\
+  \end{eqnarray*}
 </script>
   <p>Note that the above definition is deterministic because the start
-  and end times in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-852-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-852">activity</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-853-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">/</mo></mrow><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></math></span></span></span><script type="math/tex" id="MathJax-Element-853">wasStartedBy/wasEndedBy</script> statements
-  must be equal</p>.
-<p> The function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-854-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-854">time</script> mapping <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-855-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-855">Events</script> to their <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-856-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-856">Times</script> is defined
+  and end times in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-926-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-926">activity</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-927-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">/</mo></mrow><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></math></span></span></span><script type="math/tex" id="MathJax-Element-927">wasStartedBy/wasEndedBy</script> statements
+  must be equal.</p>
+<p> The function <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-928-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-928">time</script> mapping <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-929-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-929">Events</script> to their <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-930-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mi>i</mi><mi>m</mi><mi>e</mi><mi>s</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-930">Times</script> is defined
   as follows:
   </p>
-  <span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-857-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="thickmathspace" displaystyle="true" side="right" minlabelspacing="0.8em"><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mi>t</mi><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mi>t</mi><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mi>t</mi><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mi>t</mi><mtext>&nbsp;where&nbsp;</mtext><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><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><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mi>t</mi><mtext>&nbsp;where&nbsp;</mtext><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><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><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-857">\begin{eqnarray*}
+  <span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-931-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="thickmathspace" displaystyle="true" side="right" minlabelspacing="0.8em"><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mi>t</mi><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mi>t</mi><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mi>t</mi><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mi>t</mi><mtext>&nbsp;where&nbsp;</mtext><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><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><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mi>t</mi><mtext>&nbsp;where&nbsp;</mtext><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><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><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-931">\begin{eqnarray*}
   time(id) &=& t \text{ where } used(id,a,e,t,attrs) \in I\\
   time(id) &=&  t \text{ where } wasGeneratedBy(id,e,a,t,attrs) \in I\\
 time(id) &=& t\text{ where } wasInvalidatedBy(id,e,a,t,attrs) \in I\\
@@ -3784,7 +3878,7 @@
   <p>Finally, the functions giving the interpretations of the
   different identified influences are as follows:
   </p>
-<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-858-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="thickmathspace" displaystyle="true" side="right" minlabelspacing="0.8em"><mtr><mtd><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><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><mtext>&nbsp;where&nbsp;</mtext><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><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><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><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><mtext>&nbsp;where&nbsp;</mtext><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><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><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">)</mo><mtext>&nbsp;where&nbsp;</mtext><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><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><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><mi>c</mi><mi>o</mi><mi>m</mi><mi>m</mi><mi>u</mi><mi>n</mi><mi>i</mi><mi>c</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>B</mi><mi>y</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><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><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><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><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">⋅</mo><msub><mi>g</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>d</mi></mrow></msub><mo stretchy="false">⋅</mo><msub><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>d</mi></mrow></msub><mo stretchy="false">⋅</mo><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>d</mi></mrow></msub><mo stretchy="false">⋅</mo><msub><mi>e</mi><mn>1</mn></msub><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-858">\begin{eqnarray*}
+<span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-932-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right center left" rowspacing="3pt" columnspacing="thickmathspace" displaystyle="true" side="right" minlabelspacing="0.8em"><mtr><mtd><mi>u</mi><mi>s</mi><mi>e</mi><mi>d</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">,</mo><mi>e</mi><mo stretchy="false">)</mo><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mo stretchy="false">)</mo><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><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><mtext>&nbsp;where&nbsp;</mtext><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><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><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><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><mtext>&nbsp;where&nbsp;</mtext><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><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><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo stretchy="false">(</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">)</mo><mtext>&nbsp;where&nbsp;</mtext><mi>w</mi><mi>a</mi><mi>s</mi><mi>A</mi><mi>s</mi><mi>s</mi><mi>o</mi><mi>c</mi><mi>i</mi><mi>a</mi><mi>t</mi><mi>e</mi><mi>d</mi><mi>W</mi><mi>i</mi><mi>t</mi><mi>h</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">,</mo><mi>a</mi><mi>c</mi><mi>t</mi><mo stretchy="false">,</mo><mi>p</mi><mi>l</mi><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>a</mi><mi>g</mi><mo stretchy="false">)</mo><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><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><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><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><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><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><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr><mtr><mtd><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></mtd><mtd><mo stretchy="false">=</mo></mtd><mtd><msub><mi>e</mi><mn>2</mn></msub><mo stretchy="false">⋅</mo><msub><mi>g</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>d</mi></mrow></msub><mo stretchy="false">⋅</mo><msub><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>d</mi></mrow></msub><mo stretchy="false">⋅</mo><msub><mi>u</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>d</mi></mrow></msub><mo stretchy="false">⋅</mo><msub><mi>e</mi><mn>1</mn></msub><mtext>&nbsp;where&nbsp;</mtext><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><mo stretchy="false">∈</mo><mi>I</mi></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-932">\begin{eqnarray*}
   used(id) &=& (a,e) \text{ where } used(id,a,e,t,attrs) \in I\\
   generated(id) &=&  (e,a) \text{ where } wasGeneratedBy(id,e,a,t,attrs) \in I\\
   invalidated(id) &=& (e,a) \text{ where } wasInvalidatedBy(id,e,a,t,attrs) \in I\\
@@ -3795,111 +3889,152 @@
   I\\
  attributedTo(id) &=& (e,ag) \text{ where } wasAttributedTo(id,e,ag,attrs) \in I\\
 actedFor (id) &=& (ag_2,ag_1,act) \text{ where } actedOnBehalfOf(id,ag_2,ag_1,act,attrs) \in I\\
-  communicatedBy(id) &=& (a_2,a_1) \text{ where }
+  communicated(id) &=& (a_2,a_1) \text{ where }
   wasInformedBy(id,a_2,a_1,attrs)\in I\\
   derivationPath(id) &=& e_2\cdot g \cdot a \cdot u \cdot e_1 \text{ where } wasDerivedFrom(id,e_2,e_1,a,g,u,attrs) \in I\\
   derivationPath(id) &=& e_2\cdot g_{id} \cdot a_{id} \cdot u_{id} \cdot e_1 \text{ where } wasDerivedFrom(id,e_2,e_1,-,-,-,attrs) \in I\\
   \end{eqnarray*}
   </script>
-  <p>Note that since <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-859-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-859">I</script> is normalized and valid, by the uniqueness
+  <p>Note that since <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-933-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-933">I</script> is normalized and valid, by the uniqueness
   constraints these functions are all well-defined.  In the case for
   imprecise derivations, we generate additional activities, generations
-  and usages linking <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-860-Frame" style="font-size: 100%; "><span class="MathJax_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-860">e_2</script> to <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-861-Frame" style="font-size: 100%; "><span class="MathJax_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-861">e_1</script>.
+  and usages linking <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-934-Frame" style="font-size: 100%; "><span class="MathJax_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-934">e_2</script> to <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-935-Frame" style="font-size: 100%; "><span class="MathJax_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-935">e_1</script>.
   </p>
-  <div class="remark">
-
-    <p> We explicitly add activities, generations, and usages to
-  ensure that we can form derivation paths for imprecise derivation
-  formulas.
-    </p>
-    </div>
+
 </section>
 <section id="relations">
 <h4><span class="secno">6.2.3 </span>Relations</h4>
-<p>We introduced a relation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-862-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">≡</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-862">\equiv</script> corresponding to <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-863-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-863">alternateOf</script>
-  above, in defining <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-864-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-864">Things</script>, but this relation is not a component of
+<p>We introduced a relation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-936-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">≡</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-936">\equiv</script> corresponding to <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-937-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-937">alternateOf</script>
+  above, in defining <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-938-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-938">Things</script>, but this relation is not a component of
   the semantics.</p>
     <p>The event ordering relation is defined as follows:</p>
-    <span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-865-Frame" style="font-size: 100%; "><span class="MathJax_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>v</mi><mi>t</mi><mo stretchy="false">⪯</mo><mi>e</mi><mi>v</mi><msup><mi>t</mi><mo>′</mo></msup><mspace width="thickmathspace"></mspace><mo stretchy="false">⟺</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</mi><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><msub><mi>G</mi><mi>I</mi></msub></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-865">evt \preceq evt' \iff (evt,evt') \in G_I</script>
-    <p> Here, we are using a slight abuse of notation: we write <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-866-Frame" style="font-size: 100%; "><span class="MathJax_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-866">G_I</script>
-  for the directed graph that is used during validation of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-867-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-867">I</script> to
+    <span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-939-Frame" style="font-size: 100%; "><span class="MathJax_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>v</mi><mi>t</mi><mo stretchy="false">⪯</mo><mi>e</mi><mi>v</mi><msup><mi>t</mi><mo>′</mo></msup><mspace width="thickmathspace"></mspace><mo stretchy="false">⟺</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>e</mi><mi>v</mi><mi>t</mi><mo stretchy="false">,</mo><mi>e</mi><mi>v</mi><msup><mi>t</mi><mo>′</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><msub><mi>G</mi><mi>I</mi></msub></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-939">evt \preceq evt' \iff (evt,evt') \in G_I</script>
+    <p> Here, we are using a slight abuse of notation: we write <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-940-Frame" style="font-size: 100%; "><span class="MathJax_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-940">G_I</script>
+  for the directed graph that is used during validation of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-941-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-941">I</script> to
   test for cycles amond event ordering constraints.  See Sec. 7.1 of PROV-CONSTRAINTS [<cite><a class="bibref" href="#bib-PROV-CONSTRAINTS">PROV-CONSTRAINTS</a></cite>].</p>
 
-    <p> Finally, the collection membership relation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-868-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>O</mi><mi>f</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-868">MemberOf</script> is
+    <p> Finally, the collection membership relation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-942-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>O</mi><mi>f</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-942">MemberOf</script> is
     defined as follows:</p>
-    <span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-869-Frame" style="font-size: 100%; "><span class="MathJax_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"><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>O</mi><mi>f</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">⟺</mo><mspace width="thickmathspace"></mspace><mi>m</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-869">(e,c) \in MemberOf \iff memberOf(e,c) \in I</script>
+    <span class="MathJax_Preview"></span><div class="MathJax_MathML" id="MathJax-Element-943-Frame" style="font-size: 100%; "><span class="MathJax_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"><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>M</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>O</mi><mi>f</mi><mspace width="thickmathspace"></mspace><mo stretchy="false">⟺</mo><mspace width="thickmathspace"></mspace><mi>m</mi><mi>e</mi><mi>m</mi><mi>b</mi><mi>e</mi><mi>r</mi><mi>O</mi><mi>f</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">,</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">∈</mo><mi>I</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-943">(e,c) \in MemberOf \iff memberOf(e,c) \in I</script>
     
 </section>
     <section id="axioms-1">
     <h4><span class="secno">6.2.4 </span>Axioms</h4>
 
-    <p>To verify that the construction of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-870-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-870">M(I)</script> yields a PROV
+    <p>To verify that the construction of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-944-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-944">M(I)</script> yields a PROV
     structure, we must ensure that all of the axioms and
     side-conditions in the components are satisfied.  As noted above,
     the disjointness constraints are satisfied by construction. </p>
 
-    </section>
+<p>For each axiom we give the corresponding justification:</p>
+
+<ol><li>
+  Axiom 1 follows because <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-945-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-945">I</script> is normalized with respect to Inference 6.
+  </li>
+<li>
+Axioms 2 and 3 follow because <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-946-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-946">I</script> is normalized with respect to
+  Inference 9 and 10 respectively.
+  </li><li>
+Axioms 5 and 6 follow because <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-947-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-947">I</script> is normalized with respect to
+  Inference 13 and 14 respectively.
+  </li><li>
+Axioms 7 through 16 follow because <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-948-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-948">I</script> is normalized with respect to
+  Inference 15.
+  </li><li>
+Axioms 17 through 20 follow because <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-949-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-949">I</script> is normalized with respect to
+  uniqueness constraints 24 through 27.
+  </li><li>
+  Axiom 21 follows because constraints 30, 31, 33, 34 ensure that a
+  start event for an activity precedes any other start, end, usage or
+  generation events involving that activity.
+  </li>
+  <li>
+  Axiom 22 follows because constraints 30, 32, 33, 34 ensure that an
+  end event for an activity follows any other events involving that activity.
+  </li>
+  <li>
+  Axiom 23 follows because constraints 34, 36, 37, 39 ensure that a
+  generation event for an entity precedes any other events involving that entity.
+  </li>
+  <li>
+  Axiom 24 follows because constraints 36, 38, 40, 43, 44 ensure that an
+  invalidation event for an entity  follows any other generation,
+  usage, or invalidation events involving
+  that entity.
+  </li>
+  <li>Axiom 25 follows from constraint 41.</li>
+  <li> Axiom 26 follows from constraint 42 and from the fact that the
+  event ordering constraint graph <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-950-Frame" style="font-size: 100%; "><span class="MathJax_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-950">G_I</script> associated with a valid
+  instance <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-951-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-951">I</script> cannot have any cycles involving a strict precedence
+  edge.
+  </li>
+  <li> Axioms 27 through 30 follow from Constraint 47.</li>
+  <li> Axioms 31 and 32 follow from Constraint 48.</li>
+  <li> Axioms 33 and 34 follow from Constraint 49.</li>
+    
+  </ol>
+  </section>
     
 <section id="main-results">
 <h4><span class="secno">6.2.5 </span>Main results</h4>
 
 <p> The main results of this section are that if a valid PROV instance
-    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-871-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-871">I</script> normalizes to
-    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-872-Frame" style="font-size: 100%; "><span class="MathJax_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>I</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-872">I'</script>, then  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-873-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><msup><mi>I</mi><mo>′</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><mi>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-873">M(I') \models I</script>, and that furthermore <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-874-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><msup><mi>I</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-874">M(I')</script>
+    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-952-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-952">I</script> normalizes to
+    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-953-Frame" style="font-size: 100%; "><span class="MathJax_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>I</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-953">I'</script>, then  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-954-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><msup><mi>I</mi><mo>′</mo></msup><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><mi>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-954">M(I') \models I</script>, and that furthermore <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-955-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><msup><mi>I</mi><mo>′</mo></msup><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-955">M(I')</script>
     satisfies all of the inferences and constraints.  Thus, a form of
     completeness holds: every valid PROV instance has a model.</p>
 
-<div class="theorem" id="rule_40" data-count="40" data-title="Theorem 40 (rule_40)"><div class="ruleTitle"><a class="internalDFN" href="#rule_40">Theorem 40 (rule_40)</a></div>
-  <p>Suppose <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-875-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-875">J</script> is a valid PROV instance and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-876-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-876">I</script> is a normal form for
-  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-877-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-877">J</script>.  Then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-878-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-878">M(I) \models J</script></p>
+<div class="theorem" id="weak-completeness-theorem" data-count="40" data-title="Theorem 40 (weak-completeness-theorem)"><div class="ruleTitle"><a class="internalDFN" href="#weak-completeness-theorem">Theorem 40 (weak-completeness-theorem)</a></div>
+  <p>Suppose <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-956-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-956">J</script> is a valid PROV instance and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-957-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-957">I</script> is a normal form for
+  <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-958-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-958">J</script>.  Then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-959-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-959">M(I) \models J</script>.</p>
   </div>
   <div class="proof">
     <p>
-    First, we consider the case where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-879-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-879">J</script> itself is a valid,
-  normalized PROV instance <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-880-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-880">I</script>, and let <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-881-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-881">M(I)</script>
-  be the corresponding structure.  Then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-882-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-882">M(I)</script> is a PROV structure,
+    First, we consider the case where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-960-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-960">J</script> itself is a valid,
+  normalized PROV instance <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-961-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-961">I</script>, and let <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-962-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-962">M(I)</script>
+  be the corresponding structure.  Then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-963-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-963">M(I)</script> is a PROV structure,
   satisfying all of the axioms (and hence all of the inferences and
   constraints) stated above.
     </p>
 
-    <p>Moreover, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-883-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><mi>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-883">M(I) \models I</script>, as can be verified on a
+    <p>Moreover, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-964-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><mi>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-964">M(I) \models I</script>, as can be verified on a
     case-by-case basis for each type of formula by considering its
-    semantics and the definition of the construction of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-884-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-884">M</script>.  </p>
-
-    <p> To conclude, we need to show that if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-885-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-885">J</script> is not in normal
-    form, and norma,izes to <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-886-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-886">I</script>, then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-887-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-887">M(I) \models J</script>.  WE can prove
+    semantics and the definition of the construction of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-965-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-965">M</script>.  </p>
+
+    <p> To conclude, we need to show that if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-966-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-966">J</script> is not in normal
+    form, and normalizes to <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-967-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-967">I</script>, then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-968-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-968">M(I) \models J</script>.  We can prove
     this by induction on the length of the sequence of normalization
-    steps.  The base case, when <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-888-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi><mo stretchy="false">=</mo><mi>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-888">J = I</script>, is established already.
-    Suppose <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-889-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-889">J</script> normalizes in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-890-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo stretchy="false">+</mo><mn>1</mn></math></span></span></span><script type="math/tex" id="MathJax-Element-890">n+1</script> steps and we can perform one
-    normalization step on it to obtain <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-891-Frame" style="font-size: 100%; "><span class="MathJax_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>J</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-891">J'</script>, which normalizes to <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-892-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-892">I</script>
-    in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-893-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-893">n</script> steps.  By induction, we know that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-894-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><msup><mi>J</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-894">M(I) \models J'</script>.  For
-    each possible normalization step, we must show that if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-895-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><msup><mi>J</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-895">M(I)
-    \models J'</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-896-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-896">M(I) \models J</script>.
+    steps.  The base case, when <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-969-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi><mo stretchy="false">=</mo><mi>I</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-969">J = I</script>, is established already.
+    Suppose <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-970-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-970">J</script> normalizes in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-971-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo stretchy="false">+</mo><mn>1</mn></math></span></span></span><script type="math/tex" id="MathJax-Element-971">n+1</script> steps and we can perform one
+    normalization step on it to obtain <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-972-Frame" style="font-size: 100%; "><span class="MathJax_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>J</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-972">J'</script>, which normalizes to <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-973-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-973">I</script>
+    in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-974-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-974">n</script> steps.  By induction, we know that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-975-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><msup><mi>J</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-975">M(I) \models J'</script>.  For
+    each possible normalization step, we must show that if <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-976-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><msup><mi>J</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-976">M(I)
+    \models J'</script> then <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-977-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-977">M(I) \models J</script>.
     </p>
     <p>First consider inference steps.  These add information, that
-    is, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-897-Frame" style="font-size: 100%; "><span class="MathJax_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>J</mi><mo>′</mo></msup><mo stretchy="false">⊇</mo><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-897">J' \supseteq J</script>.  Hence it is immediate that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-898-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-898">M(I) \models J</script>
-    since every formula in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-899-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-899">J</script> is in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-900-Frame" style="font-size: 100%; "><span class="MathJax_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>J</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-900">J'</script>, and all formulas of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-901-Frame" style="font-size: 100%; "><span class="MathJax_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>J</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-901">J'</script>
-    are satisfied in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-902-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-902">M(I)</script>.
+    is, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-978-Frame" style="font-size: 100%; "><span class="MathJax_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>J</mi><mo>′</mo></msup><mo stretchy="false">⊇</mo><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-978">J' \supseteq J</script>.  Hence it is immediate that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-979-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-979">M(I) \models J</script>
+    since every formula in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-980-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-980">J</script> is in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-981-Frame" style="font-size: 100%; "><span class="MathJax_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>J</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-981">J'</script>, and all formulas of <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-982-Frame" style="font-size: 100%; "><span class="MathJax_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>J</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-982">J'</script>
+    are satisfied in <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-983-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-983">M(I)</script>.
     </p>
     <p>Next consider uniqueness constraint steps, which may involve
-    merging formulas.  That is, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-903-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi><mo stretchy="false">=</mo><msub><mi>J</mi><mn>0</mn></msub><mo stretchy="false">∪</mo><mo fence="false" stretchy="false">{</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-903">J = J_0 \cup
+    merging formulas.  That is, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-984-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" style="position: relative; display: inline-block; white-space: nowrap; "><span style="display: inline-block; "><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>J</mi><mo stretchy="false">=</mo><msub><mi>J</mi><mn>0</mn></msub><mo stretchy="false">∪</mo><mo fence="false" stretchy="false">{</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-984">J = J_0 \cup
     \{r(id,a_1,\ldots,a_n,attrs_1), r(id,b_1,\ldots,b_n,attrs_2)\}</script>
-    and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-904-Frame" style="font-size: 100%; "><span class="MathJax_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>J</mi><mo>′</mo></msup><mo stretchy="false">=</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>J</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∪</mo><mo fence="false" stretchy="false">{</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-904">J' = S(J_0) \cup \{r(id,S(a_1),\ldots,S(a_n),attrs_1\cup
-    attrs_2)\}</script>, where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-905-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-905">S</script> is a unifying substitution making <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-906-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><msub><mi>a</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>b</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-906">S(a_i) =
-    S(b_i)</script> for each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-907-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">∈</mo><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><mi>n</mi><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-907">i \in \{1,\ldots,n\}</script>.  Since <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-908-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><msup><mi>J</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-908">M(I) \models J'</script>,
-    we must have <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-909-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><msup><mi>J</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-909">M(I),\rho \models J'</script>  for some <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-910-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-910">\rho</script>, and therefore
-    we must also have that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-911-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>J</mi><mn>0</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-911">M(I),\rho \models S(J_0)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-912-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-912">M(I)\rho \models r(id,S(a_1),\ldots,S(a_n),attrs_1\cup
-    attrs_2)</script>.  We can extend <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-913-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-913">\rho</script> to a valuation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-914-Frame" style="font-size: 100%; "><span class="MathJax_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 mathvariant="italic">ρ</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-914">\rho'</script> such that
-    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-915-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><msup><mi mathvariant="italic">ρ</mi><mo>′</mo></msup><mo stretchy="false">⊨</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">=</mo><msub><mi>x</mi><mi>i</mi></msub><mo stretchy="false">∧</mo><mo stretchy="false">⋯</mo><mo stretchy="false">∧</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><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-915">M(I),\rho' \models S(x_1) = x_i \wedge \cdots \wedge S(x_k) =
-    x_k</script> where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-916-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>o</mi><mi>m</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><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 fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-916">dom(S) = \{x_1,\ldots,x_k\}</script>.
-    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-917-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><msup><mi mathvariant="italic">ρ</mi><mo>′</mo></msup><mo stretchy="false">⊨</mo><msub><mi>J</mi><mn>0</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-917">M(I),\rho' \models J_0</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-918-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><msup><mi mathvariant="italic">ρ</mi><mo>′</mo></msup><mo stretchy="false">⊨</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-918">M(I),\rho' \models r(id,a_1,\ldots,a_n,attrs_1\cup
-    attrs_2)</script>.  Moreover, since <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-919-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-919">S</script> is a unifier, we also have <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-920-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><msup><mi mathvariant="italic">ρ</mi><mo>′</mo></msup><mo stretchy="false">⊨</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-920">M(I),\rho' \models r(id,b_1,\ldots,b_n,attrs_1\cup
+    and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-985-Frame" style="font-size: 100%; "><span class="MathJax_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>J</mi><mo>′</mo></msup><mo stretchy="false">=</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>J</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∪</mo><mo fence="false" stretchy="false">{</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-985">J' = S(J_0) \cup \{r(id,S(a_1),\ldots,S(a_n),attrs_1\cup
+    attrs_2)\}</script>, where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-986-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-986">S</script> is a unifying substitution making <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-987-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><msub><mi>a</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>b</mi><mi>i</mi></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-987">S(a_i) =
+    S(b_i)</script> for each <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-988-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">∈</mo><mo fence="false" stretchy="false">{</mo><mn>1</mn><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><mi>n</mi><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-988">i \in \{1,\ldots,n\}</script>.  Since <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-989-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><msup><mi>J</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-989">M(I) \models J'</script>,
+    we must have <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-990-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><msup><mi>J</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-990">M(I),\rho \models J'</script>  for some <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-991-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-991">\rho</script>, and therefore
+    we must also have that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-992-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>J</mi><mn>0</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-992">M(I),\rho \models S(J_0)</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-993-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi mathvariant="italic">ρ</mi><mo stretchy="false">⊨</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-993">M(I),\rho \models r(id,S(a_1),\ldots,S(a_n),attrs_1\cup
+    attrs_2)</script>.  We can extend <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-994-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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-994">\rho</script> to a valuation <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-995-Frame" style="font-size: 100%; "><span class="MathJax_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 mathvariant="italic">ρ</mi><mo>′</mo></msup></math></span></span></span><script type="math/tex" id="MathJax-Element-995">\rho'</script> such that
+    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-996-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><msup><mi mathvariant="italic">ρ</mi><mo>′</mo></msup><mo stretchy="false">⊨</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">=</mo><msub><mi>x</mi><mn>1</mn></msub><mo stretchy="false">∧</mo><mo stretchy="false">⋯</mo><mo stretchy="false">∧</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>x</mi><mi>k</mi></msub><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-996">M(I),\rho' \models S(x_1) = x_1 \wedge \cdots \wedge S(x_k) =
+    x_k</script> where <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-997-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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>o</mi><mi>m</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo stretchy="false">=</mo><mo fence="false" stretchy="false">{</mo><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 fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-997">dom(S) = \{x_1,\ldots,x_k\}</script>.  Also, 
+    <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-998-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><msup><mi mathvariant="italic">ρ</mi><mo>′</mo></msup><mo stretchy="false">⊨</mo><msub><mi>J</mi><mn>0</mn></msub></math></span></span></span><script type="math/tex" id="MathJax-Element-998">M(I),\rho' \models J_0</script> and <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-999-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><msup><mi mathvariant="italic">ρ</mi><mo>′</mo></msup><mo stretchy="false">⊨</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-999">M(I),\rho' \models r(id,a_1,\ldots,a_n,attrs_1\cup
+    attrs_2)</script>.  Moreover, since <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-1000-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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></math></span></span></span><script type="math/tex" id="MathJax-Element-1000">S</script> is a unifier, we also have <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-1001-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><msup><mi mathvariant="italic">ρ</mi><mo>′</mo></msup><mo stretchy="false">⊨</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">∪</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-1001">M(I),\rho' \models r(id,b_1,\ldots,b_n,attrs_1\cup
     attrs_2)</script>.  Finally, since we can always remove attributes from an
-    atomic formula without danaging its satisfiability, we can
-    conclude that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-921-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><msup><mi mathvariant="italic">ρ</mi><mo>′</mo></msup><mo stretchy="false">⊨</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-921">M(I),\rho' \models r(id,a_1,\ldots,a_n,attrs_1)
+    atomic formula without damaging its satisfiability, we can
+    conclude that <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-1002-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">,</mo><msup><mi mathvariant="italic">ρ</mi><mo>′</mo></msup><mo stretchy="false">⊨</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">∧</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-1002">M(I),\rho' \models r(id,a_1,\ldots,a_n,attrs_1)
     \wedge r(id,b_1,\ldots,b_n,    attrs_2)</script>.  To conclude, we have
-    shown <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-922-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><msub><mi>J</mi><mn>0</mn></msub><mo stretchy="false">∪</mo><mo fence="false" stretchy="false">{</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-922">M(I) \models J_0 \cup \{ r(id,a_1,\ldots,a_n,attrs_1)
-   r(id,b_1,\ldots,b_n,    attrs_2)\}</script>, that is, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-923-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-923">M(I) \models J</script>, as
+    shown <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-1003-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><msub><mi>J</mi><mn>0</mn></msub><mo stretchy="false">∪</mo><mo fence="false" stretchy="false">{</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>a</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>a</mi><mi>n</mi></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mi>r</mi><mo stretchy="false">(</mo><mi>i</mi><mi>d</mi><mo stretchy="false">,</mo><msub><mi>b</mi><mn>1</mn></msub><mo stretchy="false">,</mo><mo stretchy="false">…</mo><mo stretchy="false">,</mo><msub><mi>b</mi><mi>n</mi></msub><mo stretchy="false">,</mo><mi>a</mi><mi>t</mi><mi>t</mi><mi>r</mi><msub><mi>s</mi><mn>2</mn></msub><mo stretchy="false">)</mo><mo fence="false" stretchy="false">}</mo></math></span></span></span><script type="math/tex" id="MathJax-Element-1003">M(I) \models J_0 \cup \{ r(id,a_1,\ldots,a_n,attrs_1)
+   r(id,b_1,\ldots,b_n,    attrs_2)\}</script>, that is, <span class="MathJax_Preview"></span><span class="MathJax_MathML" id="MathJax-Element-1004-Frame" style="font-size: 100%; "><span class="MathJax_MathContainer" 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><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo><mo stretchy="false">⊨</mo><mi>J</mi></math></span></span></span><script type="math/tex" id="MathJax-Element-1004">M(I) \models J</script>, as
     desired.
 </p>
 </div></section>
prov

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