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

<b>let </b><font color="Maroon" title="c1">R</font> be  <a href="scmring1.html#V1" title="SCMRING1:attr.1">good</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">c</font> be    <a href="scmring2.html#M1" title="SCMRING2:mode.1">Data-Location</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c4">i1</font> be    <a href="ami_1.html#M1" title="AMI_1:mode.1">Instruction-Location</a> of  <a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c5">s</font> be   <a href="ami_1.html#NM4" title="AMI_1:NM.4">State</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span>;<br/>









<b>reconsider </b><font color="Maroon" title="c6">I</font> = <font color="Maroon" title="c2">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Maroon" title="c4">i1</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="scmring1.html#K1" title="SCMRING1:func.1">SCM-Instr</a> <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="scmring2.html#D1" target="_self" title="SCMRING2:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">S</font> = <font color="Maroon" title="c5">s</font> as   <a href="scmring1.html#NM1" title="SCMRING1:NM.1">SCM-State</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="ref" href="scmring2.html#D1" target="_self" title="SCMRING2:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">i</font> = 7 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_1.html#K6" title="CARD_1:func.6">Segm</a> 8 <b>by </b><i><a class="ref" href="gr_cy_1.html#T10" title="GR_CY_1:th.10">GR_CY_1:10</a></i>;<br/>
<a NAME="E4:28"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c6">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i</font>,<span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">i1</font>,<font color="Maroon" title="c2">a</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
;<br/>
<a NAME="E5:28"/><i><font color="Green" title="E21">A2</font></i>: 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring1.html#K3" title="SCMRING1:func.3">IC</a> <font color="Maroon" title="c7">S</font>
 
<b>by </b><i><a class="ref" href="scmring2.html#T10" target="_self" title="SCMRING2:th.10">Th10</a></i>;<br/>
<a NAME="E6:28"/><i><font color="Green" title="E22">A3</font></i>: 
( <font color="Maroon" title="c2">a</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">SCM-Data-Loc</a>  &amp; <font color="Maroon" title="c4">i1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K6" title="AMI_2:func.6">SCM-Instr-Loc</a>  )
 
<b>by </b><i><a class="ref" href="scmring2.html#D1" target="_self" title="SCMRING2:def.1">Def1</a>, <a class="ref" href="scmring2.html#T1" target="_self" title="SCMRING2:th.1">Th1</a></i>;<br/>
<i><font color="Green" title="E23">A4</font></i>:  <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Maroon" title="c4">i1</font></span>)</span>,<font color="Maroon" title="c5">s</font> = 
 <a href="scmring1.html#K15" title="SCMRING1:func.15">SCM-Exec-Res</a> <font color="Maroon" title="c6">I</font>,<font color="Maroon" title="c7">S</font>
<b>by </b><i><a class="ref" href="scmring2.html#T12" target="_self" title="SCMRING2:th.12">Th12</a></i>
<br/>.= 
 <a href="scmring1.html#K4" title="SCMRING1:func.4">SCM-Chg</a> <font color="Maroon" title="c7">S</font>,<span class="p1">(<span class="default"><a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="scmring1.html#K6" title="SCMRING1:func.6">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="scmring1.html#K11" title="SCMRING1:func.11">cond_address</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="scmring1.html#K10" title="SCMRING1:func.10">cjump_address</a> </span>)</span>,<span class="p2">(<span class="default"><a href="ami_2.html#K18" title="AMI_2:func.18">Next</a> <span class="p3">(<span class="default"><a href="scmring1.html#K3" title="SCMRING1:func.3">IC</a> <font color="Maroon" title="c7">S</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmring2.html#E6:28"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="scmring1.html#D14" title="SCMRING1:def.14">SCMRING1:def 14</a></i>
;<br/>
<b>thus </b><a NAME="E8:28"/>
( <font color="Maroon" title="c5">s</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> implies <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Maroon" title="c4">i1</font></span>)</span>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p2">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_2"/>
<font color="Maroon" title="c5">s</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 ;<br/>

<a NAME="E2:28_2"/><b>then </b><i><font color="Green" title="E24">A5</font></i>: 
<font color="Maroon" title="c7">S</font> <a href="scmring1.html#K6" title="SCMRING1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="scmring1.html#K11" title="SCMRING1:func.11">cond_address</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="scmring2.html#E4:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="scmring2.html#E6:28"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="scmring1.html#T19" title="SCMRING1:th.19">SCMRING1:19</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Maroon" title="c4">i1</font></span>)</span>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p2">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Maroon" title="c4">i1</font></span>)</span>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
<b>by </b><i><a class="ref" href="scmring2.html#D1" target="_self" title="SCMRING2:def.1">Def1</a></i>
<br/>.= 
 <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="scmring1.html#K6" title="SCMRING1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="scmring1.html#K11" title="SCMRING1:func.11">cond_address</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="scmring1.html#K10" title="SCMRING1:func.10">cjump_address</a> </span>)</span>,<span class="p1">(<span class="default"><a href="ami_2.html#K18" title="AMI_2:func.18">Next</a> <span class="p2">(<span class="default"><a href="scmring1.html#K3" title="SCMRING1:func.3">IC</a> <font color="Maroon" title="c7">S</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmring2.html#E7:28"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="scmring1.html#T10" title="SCMRING1:th.10">SCMRING1:10</a></i>
<br/>.= 
<font color="Maroon" title="c6">I</font> <a href="scmring1.html#K10" title="SCMRING1:func.10">cjump_address</a> 
<b>by </b><i><a class="txt" href="scmring2.html#E2:28_2"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c4">i1</font>
<b>by </b><i><a class="txt" href="scmring2.html#E4:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="scmring2.html#E6:28"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="scmring1.html#T19" title="SCMRING1:th.19">SCMRING1:19</a></i>
;<br/>


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

<b>thus </b><a NAME="E9:28"/>
( <font color="Maroon" title="c5">s</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> implies <span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Maroon" title="c4">i1</font></span>)</span>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p2">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmring2.html#K11" title="SCMRING2:func.11">Next</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c5">s</font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_3"/>
<font color="Maroon" title="c5">s</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 ;<br/>

<a NAME="E2:28_3"/><b>then </b><i><font color="Green" title="E24">A6</font></i>: 
<font color="Maroon" title="c7">S</font> <a href="scmring1.html#K6" title="SCMRING1:func.6">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="scmring1.html#K11" title="SCMRING1:func.11">cond_address</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="scmring2.html#E4:28"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="scmring2.html#E6:28"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="scmring1.html#T19" title="SCMRING1:th.19">SCMRING1:19</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Maroon" title="c4">i1</font></span>)</span>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p2">(<span class="default"><a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Maroon" title="c4">i1</font></span>)</span>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
<b>by </b><i><a class="ref" href="scmring2.html#D1" target="_self" title="SCMRING2:def.1">Def1</a></i>
<br/>.= 
 <a href="funcop_1.html#K13" title="FUNCOP_1:func.13">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="scmring1.html#K6" title="SCMRING1:func.6">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="scmring1.html#K11" title="SCMRING1:func.11">cond_address</a> </span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">I</font> <a href="scmring1.html#K10" title="SCMRING1:func.10">cjump_address</a> </span>)</span>,<span class="p1">(<span class="default"><a href="ami_2.html#K18" title="AMI_2:func.18">Next</a> <span class="p2">(<span class="default"><a href="scmring1.html#K3" title="SCMRING1:func.3">IC</a> <font color="Maroon" title="c7">S</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scmring2.html#E7:28"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="scmring1.html#T10" title="SCMRING1:th.10">SCMRING1:10</a></i>
<br/>.= 
 <a href="ami_2.html#K18" title="AMI_2:func.18">Next</a> <span class="p1">(<span class="default"><a href="scmring1.html#K3" title="SCMRING1:func.3">IC</a> <font color="Maroon" title="c7">S</font></span>)</span>
<b>by </b><i><a class="txt" href="scmring2.html#E2:28_3"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
<br/>.= 
 <a href="scmring2.html#K11" title="SCMRING2:func.11">Next</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c5">s</font></span>)</span>
<b>by </b><i><a class="txt" href="scmring2.html#E5:28"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="scmring2.html#T11" target="_self" title="SCMRING2:th.11">Th11</a></i>
;<br/>


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

<a NAME="E10:28"/>
<font color="Maroon" title="c3">c</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">SCM-Data-Loc</a> 
 
<b>by </b><i><a class="ref" href="scmring2.html#T1" target="_self" title="SCMRING2:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E11:28"/>
<span class="p1">(<span class="default"><a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="scmring2.html#K10" title="SCMRING2:func.10">=0_goto</a> <font color="Maroon" title="c4">i1</font></span>)</span>,<font color="Maroon" title="c5">s</font></span>)</span> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> <a href="scmring2.html#K3" title="SCMRING2:func.3">.</a> <font color="Maroon" title="c3">c</font>
 <b>by </b><i><a class="txt" href="scmring2.html#E7:28"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="scmring1.html#T11" title="SCMRING1:th.11">SCMRING1:11</a></i>;<br/>


</div>
