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<span class="kw">let </span><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</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">  for <font color="Olive" title="b1">s1</font>, <font color="Olive" title="b2">s2</font> being   <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><br/>  for <font color="Olive" title="b3">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> <br/>  for <font color="Olive" title="b4">I</font> being   <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>  st  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Olive" title="b1">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Olive" title="b2">s2</font> &amp;  <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">I</font>,<font color="Olive" title="b1">s1</font>,<font color="Maroon" title="c3">p1</font> holds <br/> <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">I</font>,<font color="Olive" title="b2">s2</font>,<font color="Maroon" title="c4">p2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">s1</font>, <font color="Maroon" title="c6">s2</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">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> <br/>  for <font color="Olive" title="b2">I</font> being   <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>  st  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Maroon" title="c6">s2</font> &amp;  <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">I</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c3">p1</font> holds <br/> <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">I</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c4">p2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">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">  for <font color="Olive" title="b1">I</font> being   <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>  st  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Maroon" title="c6">s2</font> &amp;  <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Maroon" title="c7">a</font>,<font color="Olive" title="b1">I</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c3">p1</font> holds <br/> <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Maroon" title="c7">a</font>,<font color="Olive" title="b1">I</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c4">p2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">I</font> be   <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"> (  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Maroon" title="c6">s2</font> &amp;  <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c3">p1</font> implies  <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c4">p2</font> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:47"/><span class="lab"><font color="Green" title="E41">A1</font></span>: 
 <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Maroon" title="c5">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <font color="Maroon" title="c6">s2</font>
 <span class="kw">and </span><br/><a NAME="E2:47"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
 <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c5">s1</font>,<font color="Maroon" title="c3">p1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="scmfsa9a.html#R3" title="SCMFSA9A:pred.3">ProperBodyWhile&gt;0</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c6">s2</font>,<font color="Maroon" title="c4">p2</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">k</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> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="scmfsa9a.html#D4" target="_self" title="SCMFSA9A:def.4">SCMFSA9A:def 4</a> <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"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">s2</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c7">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>  implies ( <font color="Maroon" title="c8">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">s2</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c4">p2</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>)</span>)</span> &amp; <font color="Maroon" title="c8">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">s2</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c4">p2</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>)</span>)</span> ) )</span><br/>

<span class="kw">assume </span><a NAME="E3:47"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">s2</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c7">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"> ( <font color="Maroon" title="c8">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">s2</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c4">p2</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>)</span>)</span> &amp; <font color="Maroon" title="c8">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">s2</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c4">p2</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>)</span>)</span> )</span><br/>

<a NAME="E4:47"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
 <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">s2</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">DataPart</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">s1</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:47"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="scmfsa9a.html#T34" target="_self" title="SCMFSA9A:th.34">Th40</a></span>;<br/>
<a NAME="E5:47"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">s1</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="scmfsa_2.html#K18" title="SCMFSA_2:func.18">.</a> <font color="Maroon" title="c7">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="txt" href="#E3:47"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="scmfsa6a.html#T7" title="SCMFSA6A:th.7">SCMFSA6A:7</a></span>;<br/>
<a NAME="E6:47"/><span class="kw">then </span>
( <font color="Maroon" title="c8">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">s1</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c3">p1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>)</span>)</span> &amp; <font color="Maroon" title="c8">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">s1</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c3">p1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:47"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="scmfsa9a.html#D4" target="_self" title="SCMFSA9A:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:47"/>
( <font color="Maroon" title="c8">I</font> <a href="scmfsa7b.html#R5" title="SCMFSA7B:pred.5">is_closed_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">s2</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c4">p2</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>)</span>)</span> &amp; <font color="Maroon" title="c8">I</font> <a href="scmfsa7b.html#R6" title="SCMFSA7B:pred.6">is_halting_on</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K5" title="SCMFSA_9:func.5">StepWhile&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">s2</font>)</span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c4">p2</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="scmfsa_9.html#K2" title="SCMFSA_9:func.2">while&gt;0</a> (<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">I</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:47"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="scmfsa8b.html#T5" title="SCMFSA8B:th.5">SCMFSA8B:5</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>
