<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">I</font> be   <a href="scmfsa7b.html#V1" title="SCMFSA7B:attr.1">good</a>   <a href="amistd_1.html#V7" title="AMISTD_1:attr.7">parahalting</a>  <a href="compos_1.html#NM5" title="COMPOS_1:NM.5">Program</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</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">a</font> being   <a href="sf_mastr.html#NV2" title="SF_MASTR:NV.2">read-write</a>   <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a>   st  not <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R4" title="SCMFSA7B:pred.4">destroys</a> <font color="Olive" title="b1">a</font> holds <br/> <a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 1</span>)</span> is  <a href="scmfsa8c.html#K2" title="SCMFSA8C:func.2">Times</a> (<font color="Olive" title="b1">a</font>,<font color="Maroon" title="c3">I</font>) <a href="extpro_1.html#V5" title="EXTPRO_1:attr.5">-halted</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">a</font> be   <a href="sf_mastr.html#NV2" title="SF_MASTR:NV.2">read-write</a>   <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R4" title="SCMFSA7B:pred.4">destroys</a> <font color="Maroon" title="c4">a</font> implies  <a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 1</span>)</span> is  <a href="scmfsa8c.html#K2" title="SCMFSA8C:func.2">Times</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="extpro_1.html#V5" title="EXTPRO_1:attr.5">-halted</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:87"/><span class="lab"><font color="Green" title="E66">A1</font></span>: 
 not <font color="Maroon" title="c3">I</font> <a href="scmfsa7b.html#R4" title="SCMFSA7B:pred.4">destroys</a> <font color="Maroon" title="c4">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 1</span>)</span> is  <a href="scmfsa8c.html#K2" title="SCMFSA8C:func.2">Times</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="extpro_1.html#V5" title="EXTPRO_1:attr.5">-halted</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">s</font> be   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</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">P</font> being   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> holds   <a href="scmfsa8c.html#K2" title="SCMFSA8C:func.2">Times</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a>  <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">P</font> be   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="scmfsa8c.html#K2" title="SCMFSA8C:func.2">Times</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a>  <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:87_1_1"/>
( <font color="Maroon" title="c5">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c5">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:87_1_1_1"/>
<font color="Maroon" title="c5">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="scmfsa8c.html#K2" title="SCMFSA8C:func.2">Times</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a>  <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><a NAME="E2:87_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A2</font></span>: 
<span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa6c.html#T3" title="SCMFSA6C:th.3">SCMFSA6C:3</a></span>;<br/><a NAME="E3:87_1_1_1"/><span class="lab"><font color="Green" title="E68">A3</font></span>: 
 <a href="scmfsa6a.html#K2" title="SCMFSA6A:func.2">Directed</a> <span class="p1">(<span class="default"><a href="scmfsa8c.html#K1" title="SCMFSA8C:func.1">loop</a> <span class="p2">(<span class="default"><a href="scmfsa8b.html#K1" title="SCMFSA8B:func.1">if=0</a> (<font color="Maroon" title="c4">a</font>,<span class="p3">(<span class="default"><a href="scmfsa8a.html#K1" title="SCMFSA8A:func.1">Goto</a> 2</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="scmfsa6a.html#K6" title="SCMFSA6A:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">SubFrom</a> (<font color="Maroon" title="c4">a</font>,<span class="p5">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>)</span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa8c.html#K1" title="SCMFSA8C:func.1">loop</a> <span class="p1">(<span class="default"><a href="scmfsa8b.html#K1" title="SCMFSA8B:func.1">if=0</a> (<font color="Maroon" title="c4">a</font>,<span class="p2">(<span class="default"><a href="scmfsa8a.html#K1" title="SCMFSA8A:func.1">Goto</a> 2</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="scmfsa6a.html#K6" title="SCMFSA6A:func.6">';'</a> <span class="p3">(<span class="default"><a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">SubFrom</a> (<font color="Maroon" title="c4">a</font>,<span class="p4">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>)</span>)</span></span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa6a.html#T22" title="SCMFSA6A:th.22">SCMFSA6A:22</a></span>;<br/><a NAME="E4:87_1_1_1"/>
<span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa6a.html#T38" title="SCMFSA6A:th.38">SCMFSA6A:38</a></span>;<br/><a NAME="E5:87_1_1_1"/><span class="kw">then </span>
 <a href="scmfsa6a.html#K2" title="SCMFSA6A:func.2">Directed</a> <span class="p1">(<span class="default"><a href="scmfsa8c.html#K1" title="SCMFSA8C:func.1">loop</a> <span class="p2">(<span class="default"><a href="scmfsa8b.html#K1" title="SCMFSA8B:func.1">if=0</a> (<font color="Maroon" title="c4">a</font>,<span class="p3">(<span class="default"><a href="scmfsa8a.html#K1" title="SCMFSA8A:func.1">Goto</a> 2</span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="scmfsa6a.html#K6" title="SCMFSA6A:func.6">';'</a> <span class="p4">(<span class="default"><a href="scmfsa_2.html#K8" title="SCMFSA_2:func.8">SubFrom</a> (<font color="Maroon" title="c4">a</font>,<span class="p5">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>)</span>)</span></span>)</span>)</span>)</span></span>)</span> <a href="scmfsa8a.html#R1" title="SCMFSA8A:pred.1">is_pseudo-closed_on</a>  <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="txt" href="#E2:87_1_1_1"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="txt" href="#E3:87_1_1_1"><span class="lab"><font color="Green" title="E68">A3</font></span></a>, <a class="ref" href="scmfsa8c.html#T84" target="_self" title="SCMFSA8C:th.84">Th117</a></span>;<br/><span class="kw">hence </span><a NAME="E6:87_1_1_1"/>
 <a href="scmfsa8c.html#K2" title="SCMFSA8C:func.2">Times</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a>  <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87_1_1_1"><span class="lab"><font color="Green" title="E67">A2</font></span></a>, <a class="ref" href="scmfsa8c.html#T39" target="_self" title="SCMFSA8C:th.39">Th68</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="87_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:87_1_1_2"/><span class="lab"><font color="Green" title="E67">A4</font></span>: 
<font color="Maroon" title="c5">s</font> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="scmfsa8c.html#K2" title="SCMFSA8C:func.2">Times</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a>  <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><a NAME="E2:87_1_1_2"/><span class="lab"><font color="Green" title="E68">A5</font></span>: 
 <a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a>  <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa7b.html#T18" title="SCMFSA7B:th.18">SCMFSA7B:18</a></span>;<br/><a NAME="E3:87_1_1_2"/><span class="lab"><font color="Green" title="E69">A6</font></span>: 
 <a href="compos_1.html#K6" title="COMPOS_1:func.6">Stop</a> <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a>  <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa7b.html#T19" title="SCMFSA7B:th.19">SCMFSA7B:19</a></span>;<br/><a NAME="E4:87_1_1_2"/>
<span class="p1">(<span class="default"><a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c4">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87_1_1_2"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="ref" href="scmfsa6c.html#T3" title="SCMFSA6C:th.3">SCMFSA6C:3</a></span>;<br/><span class="kw">hence </span><a NAME="E5:87_1_1_2"/>
 <a href="scmfsa8c.html#K2" title="SCMFSA8C:func.2">Times</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a>  <a href="scmfsa6a.html#K3" title="SCMFSA6A:func.3">Initialized</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c6">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87_1_1_2"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="txt" href="#E3:87_1_1_2"><span class="lab"><font color="Green" title="E69">A6</font></span></a>, <a class="ref" href="scmfsa8b.html#T21" title="SCMFSA8B:th.21">SCMFSA8B:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:87"/>
 <a href="memstr_0.html#K6" title="MEMSTR_0:func.6">Initialize</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_2.html#K4" title="SCMFSA_2:func.4">intloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> 1</span>)</span> is  <a href="scmfsa8c.html#K2" title="SCMFSA8C:func.2">Times</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">I</font>) <a href="extpro_1.html#V5" title="EXTPRO_1:attr.5">-halted</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmfsa8c.html#T6" target="_self" title="SCMFSA8C:th.6">Th24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
