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<div class="add">

<b>let </b><font color="Maroon" title="c1">PTN</font> be    <a href="petri.html#L1" title="PETRI:struct.1">PT_net_Str</a> ;<br/><b>let </b><font color="Maroon" title="c2">Sd</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="petri.html#U1" title="PETRI:sel.1">Places</a> of <font color="Maroon" title="c1">PTN</font>;<br/>



<b>thus </b><a NAME="E1:15"/>
( <font color="Maroon" title="c2">Sd</font> is <a href="petri.html#V2" title="PETRI:attr.2">Deadlock-like</a> implies for <font color="Olive" title="b1">M0</font> being  <a href="boolmark.html#NM1" title="BOOLMARK:NM.1">Boolean_marking</a> of <font color="Maroon" title="c1">PTN</font>  st <font color="Olive" title="b1">M0</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c2">Sd</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/>for <font color="Olive" title="b2">t</font> being  <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c1">PTN</font>  st <font color="Olive" title="b2">t</font> <a href="boolmark.html#R1" title="BOOLMARK:pred.1">is_firable_on</a> <font color="Olive" title="b1">M0</font> holds <br/><span class="p1">(<span class="default"><a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">M0</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c2">Sd</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon" title="c2">Sd</font> is <a href="petri.html#V2" title="PETRI:attr.2">Deadlock-like</a>
 ;<br/>

<a NAME="E2:15_1"/><b>then </b><i><font color="Green" title="E10">A1</font></i>: 
 <a href="petri.html#K5" title="PETRI:func.5">*'</a> <font color="Maroon" title="c2">Sd</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">Sd</font> <a href="petri.html#K6" title="PETRI:func.6">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="petri.html#D5" title="PETRI:def.5">PETRI:def 5</a></i>;<br/>
<b>let </b><font color="Maroon" title="c3">M0</font> be   <a href="boolmark.html#NM1" title="BOOLMARK:NM.1">Boolean_marking</a> of <font color="Maroon" title="c1">PTN</font>;<br/>

<b>assume </b><a NAME="E3:15_1"/><i><font color="Green" title="E11">A2</font></i>: 
<font color="Maroon" title="c3">M0</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c2">Sd</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/>

<b>let </b><font color="Maroon" title="c4">t</font> be   <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c1">PTN</font>;<br/>

<b>assume </b><a NAME="E4:15_1"/>
<font color="Maroon" title="c4">t</font> <a href="boolmark.html#R1" title="BOOLMARK:pred.1">is_firable_on</a> <font color="Maroon" title="c3">M0</font>
 ;<br/>

<a NAME="E5:15_1"/><b>then </b>
<font color="Maroon" title="c3">M0</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="petri.html#K7" title="PETRI:func.7">*'</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="boolmark.html#D2" target="_self" title="BOOLMARK:def.2">Def2</a></i>;<br/>
<a NAME="E6:15_1"/><b>then </b><i><font color="Green" title="E12">A3</font></i>: 
<font color="Maroon" title="c3">M0</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1">(<span class="default"><a href="petri.html#K7" title="PETRI:func.7">*'</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a>, <a class="ref" href="zfmisc_1.html#T17" title="ZFMISC_1:th.17">ZFMISC_1:17</a></i>;<br/>
<a NAME="E7:15_1"/><b>then </b>
 <a href="petri.html#K7" title="PETRI:func.7">*'</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">Sd</font>
 
<b>by </b><i><a class="txt" href="boolmark.html#E3:15_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="boolmark.html#T2" target="_self" title="BOOLMARK:th.2">Th2</a></i>;<br/>
<a NAME="E8:15_1"/><b>then </b>
not <font color="Maroon" title="c4">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="petri.html#K5" title="PETRI:func.5">*'</a> <font color="Maroon" title="c2">Sd</font>
 
<b>by </b><i><a class="txt" href="boolmark.html#E2:15_1"><i><font color="Green" title="E10">A1</font></i></a>, <a class="ref" href="petri.html#T19" title="PETRI:th.19">PETRI:19</a></i>;<br/>
<a NAME="E9:15_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="petri.html#K8" title="PETRI:func.8">*'</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">Sd</font>
 
<b>by </b><i><a class="ref" href="petri.html#T20" title="PETRI:th.20">PETRI:20</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <font color="Maroon" title="c4">t</font>,<font color="Maroon" title="c3">M0</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c2">Sd</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">M0</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="petri.html#K7" title="PETRI:func.7">*'</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">t</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c2">Sd</font>
<b>by </b><i><a class="ref" href="boolmark.html#T3" target="_self" title="BOOLMARK:th.3">Th3</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="txt" href="boolmark.html#E3:15_1"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="boolmark.html#E6:15_1"><i><font color="Green" title="E12">A3</font></i></a>, <a class="ref" href="boolmark.html#T2" target="_self" title="BOOLMARK:th.2">Th2</a>, <a class="ref" href="boolmark.html#T3" target="_self" title="BOOLMARK:th.3">Th3</a></i>
;<br/><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:15"/>
( ( for <font color="Olive" title="b1">M0</font> being  <a href="boolmark.html#NM1" title="BOOLMARK:NM.1">Boolean_marking</a> of <font color="Maroon" title="c1">PTN</font>  st <font color="Olive" title="b1">M0</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c2">Sd</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> holds <br/>for <font color="Olive" title="b2">t</font> being  <a href="petri.html#NM4" title="PETRI:NM.4">transition</a> of <font color="Maroon" title="c1">PTN</font>  st <font color="Olive" title="b2">t</font> <a href="boolmark.html#R1" title="BOOLMARK:pred.1">is_firable_on</a> <font color="Olive" title="b1">M0</font> holds <br/><span class="p1">(<span class="default"><a href="boolmark.html#K2" title="BOOLMARK:func.2">Firing</a> <font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">M0</font></span>)</span> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c2">Sd</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) implies <font color="Maroon" title="c2">Sd</font> is <a href="petri.html#V2" title="PETRI:attr.2">Deadlock-like</a> )
 <b>by </b><i><a class="txt" href="boolmark.html#E12"><i><font color="Green" title="E9">Lm1</font></i></a></i>;<br/>


</div>
