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<span class="kw">let </span><font color="Maroon" title="c1">P</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being   <a href="pnproc_1.html#NM1" title="PNPROC_1:NM.1">marking</a> of <font color="Maroon" title="c1">P</font><br/>  for <font color="Olive" title="b2">t1</font>, <font color="Olive" title="b3">t2</font> being    <a href="pnproc_1.html#M1" title="PNPROC_1:mode.1">transition</a> of <font color="Maroon" title="c1">P</font>  st <span class="p1">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Olive" title="b2">t1</font></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Olive" title="b3">t2</font></span>)</span> <a href="pnproc_1.html#R2" title="PNPROC_1:pred.2">c=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="pnproc_1.html#K6" title="PNPROC_1:func.6">fire</a> (<font color="Olive" title="b3">t2</font>,<span class="p1">(<span class="default"><a href="pnproc_1.html#K6" title="PNPROC_1:func.6">fire</a> (<font color="Olive" title="b2">t1</font>,<font color="Olive" title="b1">m</font>)</span>)</span>) <a href="pnproc_1.html#R1" title="PNPROC_1:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">m</font> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <span class="p4">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Olive" title="b2">t1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <span class="p3">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Olive" title="b3">t2</font></span>)</span></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p2">(<span class="default"><a href="pnproc_1.html#K5" title="PNPROC_1:func.5">Post</a> <font color="Olive" title="b2">t1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K5" title="PNPROC_1:func.5">Post</a> <font color="Olive" title="b3">t2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">m</font> be   <a href="pnproc_1.html#NM1" title="PNPROC_1:NM.1">marking</a> of <font color="Maroon" title="c1">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">t1</font>, <font color="Olive" title="b2">t2</font> being    <a href="pnproc_1.html#M1" title="PNPROC_1:mode.1">transition</a> of <font color="Maroon" title="c1">P</font>  st <span class="p1">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Olive" title="b1">t1</font></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Olive" title="b2">t2</font></span>)</span> <a href="pnproc_1.html#R2" title="PNPROC_1:pred.2">c=</a> <font color="Maroon" title="c2">m</font> holds <br/> <a href="pnproc_1.html#K6" title="PNPROC_1:func.6">fire</a> (<font color="Olive" title="b2">t2</font>,<span class="p1">(<span class="default"><a href="pnproc_1.html#K6" title="PNPROC_1:func.6">fire</a> (<font color="Olive" title="b1">t1</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>) <a href="pnproc_1.html#R1" title="PNPROC_1:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <span class="p4">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Olive" title="b1">t1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <span class="p3">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Olive" title="b2">t2</font></span>)</span></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p2">(<span class="default"><a href="pnproc_1.html#K5" title="PNPROC_1:func.5">Post</a> <font color="Olive" title="b1">t1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K5" title="PNPROC_1:func.5">Post</a> <font color="Olive" title="b2">t2</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">t1</font>, <font color="Maroon" title="c4">t2</font> be    <a href="pnproc_1.html#M1" title="PNPROC_1:mode.1">transition</a> of <font color="Maroon" title="c1">P</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c3">t1</font></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c4">t2</font></span>)</span> <a href="pnproc_1.html#R2" title="PNPROC_1:pred.2">c=</a> <font color="Maroon" title="c2">m</font> implies  <a href="pnproc_1.html#K6" title="PNPROC_1:func.6">fire</a> (<font color="Maroon" title="c4">t2</font>,<span class="p1">(<span class="default"><a href="pnproc_1.html#K6" title="PNPROC_1:func.6">fire</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>) <a href="pnproc_1.html#R1" title="PNPROC_1:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <span class="p4">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c3">t1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <span class="p3">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c4">t2</font></span>)</span></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p2">(<span class="default"><a href="pnproc_1.html#K5" title="PNPROC_1:func.5">Post</a> <font color="Maroon" title="c3">t1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K5" title="PNPROC_1:func.5">Post</a> <font color="Maroon" title="c4">t2</font></span>)</span> )</span><br/>







<span class="kw">assume </span><a NAME="E1:48"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<span class="p1">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c3">t1</font></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c4">t2</font></span>)</span> <a href="pnproc_1.html#R2" title="PNPROC_1:pred.2">c=</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="pnproc_1.html#K6" title="PNPROC_1:func.6">fire</a> (<font color="Maroon" title="c4">t2</font>,<span class="p1">(<span class="default"><a href="pnproc_1.html#K6" title="PNPROC_1:func.6">fire</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>) <a href="pnproc_1.html#R1" title="PNPROC_1:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <span class="p4">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c3">t1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <span class="p3">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c4">t2</font></span>)</span></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p2">(<span class="default"><a href="pnproc_1.html#K5" title="PNPROC_1:func.5">Post</a> <font color="Maroon" title="c3">t1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K5" title="PNPROC_1:func.5">Post</a> <font color="Maroon" title="c4">t2</font></span>)</span></span><br/>

<a NAME="E2:48"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
 <a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c3">t1</font> <a href="pnproc_1.html#R2" title="PNPROC_1:pred.2">c=</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c3">t1</font></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c4">t2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pnproc_1.html#T15" target="_self" title="PNPROC_1:th.15">Th16</a></span>;<br/>
<a NAME="E3:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A3</font></span>: 
 <a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c3">t1</font> <a href="pnproc_1.html#R2" title="PNPROC_1:pred.2">c=</a> <font color="Maroon" title="c2">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="pnproc_1.html#T13" target="_self" title="PNPROC_1:th.13">Th14</a></span>;<br/>
<a NAME="E4:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 <a href="pnproc_1.html#K6" title="PNPROC_1:func.6">fire</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c2">m</font>) <a href="pnproc_1.html#R1" title="PNPROC_1:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <span class="p2">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c3">t1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K5" title="PNPROC_1:func.5">Post</a> <font color="Maroon" title="c3">t1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pnproc_1.html#D7" target="_self" title="PNPROC_1:def.7">Def8</a></span>;<br/>
<a NAME="E5:48"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
 <a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c4">t2</font> <a href="pnproc_1.html#R1" title="PNPROC_1:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c4">t2</font></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p2">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c3">t1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c3">t1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pnproc_1.html#T18" target="_self" title="PNPROC_1:th.18">Th19</a></span>;<br/>
<a NAME="E6:48"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c3">t1</font></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p2">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c4">t2</font></span>)</span></span>)</span> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c3">t1</font></span>)</span> <a href="pnproc_1.html#R2" title="PNPROC_1:pred.2">c=</a> <font color="Maroon" title="c2">m</font> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c3">t1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="pnproc_1.html#T19" target="_self" title="PNPROC_1:th.19">Th20</a></span>;<br/>
<a NAME="E7:48"/>
<font color="Maroon" title="c2">m</font> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c3">t1</font></span>)</span> <a href="pnproc_1.html#R2" title="PNPROC_1:pred.2">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <span class="p2">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c3">t1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K5" title="PNPROC_1:func.5">Post</a> <font color="Maroon" title="c3">t1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pnproc_1.html#T15" target="_self" title="PNPROC_1:th.15">Th16</a></span>;<br/>
<a NAME="E8:48"/><span class="kw">then </span>
 <a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c4">t2</font> <a href="pnproc_1.html#R2" title="PNPROC_1:pred.2">c=</a>  <a href="pnproc_1.html#K6" title="PNPROC_1:func.6">fire</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c2">m</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E5:48"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="pnproc_1.html#T13" target="_self" title="PNPROC_1:th.13">Th14</a></span>;<br/>
<span class="kw">hence </span> <a href="pnproc_1.html#K6" title="PNPROC_1:func.6">fire</a> (<font color="Maroon" title="c4">t2</font>,<span class="p1">(<span class="default"><a href="pnproc_1.html#K6" title="PNPROC_1:func.6">fire</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c2">m</font>)</span>)</span>) = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pnproc_1.html#K6" title="PNPROC_1:func.6">fire</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c2">m</font>)</span>)</span> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <span class="p2">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c4">t2</font></span>)</span></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K5" title="PNPROC_1:func.5">Post</a> <font color="Maroon" title="c4">t2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pnproc_1.html#D7" target="_self" title="PNPROC_1:def.7">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <span class="p4">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c3">t1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p3">(<span class="default"><a href="pnproc_1.html#K5" title="PNPROC_1:func.5">Post</a> <font color="Maroon" title="c3">t1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <span class="p2">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c4">t2</font></span>)</span></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K5" title="PNPROC_1:func.5">Post</a> <font color="Maroon" title="c4">t2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="pnproc_1.html#D7" target="_self" title="PNPROC_1:def.7">Def8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <span class="p4">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c3">t1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K3" title="PNPROC_1:func.3">-</a> <span class="p3">(<span class="default"><a href="pnproc_1.html#K4" title="PNPROC_1:func.4">Pre</a> <font color="Maroon" title="c4">t2</font></span>)</span></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p2">(<span class="default"><a href="pnproc_1.html#K5" title="PNPROC_1:func.5">Post</a> <font color="Maroon" title="c3">t1</font></span>)</span></span>)</span> <a href="pnproc_1.html#K2" title="PNPROC_1:func.2">+</a> <span class="p1">(<span class="default"><a href="pnproc_1.html#K5" title="PNPROC_1:func.5">Post</a> <font color="Maroon" title="c4">t2</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E5:48"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="pnproc_1.html#T19" target="_self" title="PNPROC_1:th.19">Th20</a>, <a class="ref" href="pnproc_1.html#T20" target="_self" title="PNPROC_1:th.20">Th21</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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