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<span class="kw">let </span><font color="Maroon" title="c2">I</font> be   <a href="compos_1.html#NM5" title="COMPOS_1:NM.5">Program</a> of ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">I</font> is  <a href="amistd_1.html#V7" title="AMISTD_1:attr.7">parahalting</a>  implies <font color="Maroon" title="c2">I</font> is  <a href="amistd_1.html#V6" title="AMISTD_1:attr.6">paraclosed</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:7_1_1"/><span class="lab"><font color="Green" title="E5">Z</font></span>: 
<font color="Maroon" title="c2">I</font> is  <a href="amistd_1.html#V7" title="AMISTD_1:attr.7">parahalting</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">I</font> is  <a href="amistd_1.html#V6" title="AMISTD_1:attr.6">paraclosed</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">s</font> be   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="memstr_0.html#V4" title="MEMSTR_0:attr.4">-started</a>  <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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="amistd_1.html#D10" title="AMISTD_1:def.10">AMISTD_1:def 10</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">b<sub>1</sub></font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive">b<sub>1</sub></font> or  for <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c3">s</font>,<font color="Olive">b<sub>2</sub></font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relset_1.html#K1" title="RELSET_1:func.1">K325</a>(<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<font color="Maroon" title="c2">I</font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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"> (  not <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P</font> or  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Olive">b<sub>1</sub></font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relset_1.html#K1" title="RELSET_1:func.1">K325</a>(<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<font color="Maroon" title="c2">I</font>) )</span><br/>



<span class="kw">assume </span><a NAME="E2:7_1_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">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">b<sub>1</sub></font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Olive">b<sub>1</sub></font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relset_1.html#K1" title="RELSET_1:func.1">K325</a>(<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<font color="Maroon" title="c2">I</font>)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <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#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c5">n</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="relset_1.html#K1" title="RELSET_1:func.1">K325</a>(<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<font color="Maroon" title="c2">I</font>)</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> not  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon">c<sub>1</sub></font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font>;<br/>
<span class="kw">assume </span><a NAME="E3:7_1_1"/>
 not  <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c5">n</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A4</font></span>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">n</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:7_1_1"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">n</font>]
 <span class="kw">and </span><br/><a NAME="E7:7_1_1"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E4:7_1_1"><span class="lab"><font color="Green" title="E7">A4</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c7">n</font> = <font color="Maroon" title="c6">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E9:7_1_1"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">n</font> holds <br/> <a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Olive" title="b1">m</font>)</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">I</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:7_1_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">s2</font> =  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c7">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c9">s0</font> = <font color="Maroon" title="c3">s</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">s1</font> =  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c7">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c11">P0</font> = <font color="Maroon" title="c4">P</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p1">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p2">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c7">n</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <span class="p2">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c7">n</font>)</span>)</span></span>)</span></span>)</span>);<br/>
<a NAME="E10:7_1_1"/><span class="lab"><font color="Green" title="E11">A13</font></span>: 
<font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">P</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p1">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p2">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c7">n</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <span class="p2">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c7">n</font>)</span>)</span></span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:7_1_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E2:7_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="funct_7.html#T89" title="FUNCT_7:th.89">FUNCT_7:89</a></span>;<br/>
<a NAME="E11:7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">B14</font></span>: 
 <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p2">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c7">n</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <span class="p3">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p4">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c7">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c7">n</font>) <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c7">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:7_1_1"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E2:7_1_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="amistd_2.html#T10" title="AMISTD_2:th.10">AMISTD_2:10</a></span>;<br/>
<a NAME="E12:7_1_1"/><span class="lab"><font color="Green" title="E13">B15</font></span>: 
 not <font color="Maroon" title="c4">P</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p1">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p2">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c7">n</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <span class="p2">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c7">n</font>)</span>)</span></span>)</span></span>)</span>) <a href="extpro_1.html#R1" title="EXTPRO_1:pred.1">halts_on</a>  <a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p2">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c7">n</font>)</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <span class="p3">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p4">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c7">n</font>)</span>)</span></span>)</span></span>)</span>)</span>)</span>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c7">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:7_1_1"><span class="lab"><font color="Green" title="E12">B14</font></span></a>, <a class="txt" href="scmfsa6b.html#E7"><span class="lab"><font color="Green" title="E4">Lm45</font></span></a></span>;<br/>
<a NAME="E13:7_1_1"/>
<font color="Maroon" title="c4">P</font> <a href="funct_7.html#K2" title="FUNCT_7:func.2">+*</a> (<span class="p1">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p2">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c7">n</font>)</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="scmfsa_2.html#K11" title="SCMFSA_2:func.11">goto</a> <span class="p2">(<span class="default"><a href="memstr_0.html#K3" title="MEMSTR_0:func.3">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K5" title="EXTPRO_1:func.5">Comput</a> (<font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c3">s</font>,<font color="Maroon" title="c7">n</font>)</span>)</span></span>)</span></span>)</span>) <a href="extpro_1.html#R1" title="EXTPRO_1:pred.1">halts_on</a> <font color="Maroon" title="c3">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:7_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E1:7_1_1"><span class="lab"><font color="Green" title="E5">Z</font></span></a>, <a class="ref" href="amistd_1.html#D11" title="AMISTD_1:def.11">AMISTD_1:def 11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:7_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:7_1_1"><span class="lab"><font color="Green" title="E13">B15</font></span></a>, <a class="ref" href="extpro_1.html#T22" title="EXTPRO_1:th.22">EXTPRO_1:22</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>
