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<span class="kw">let </span><font color="Maroon" title="c6">s</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">k1</font>, <font color="Olive" title="b2">k2</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/> for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>  holds <br/> ( ( <font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">a</font></span>)</span>,<font color="Olive" title="b1">k1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  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="Olive" title="b3">a</font>,<font color="Olive" title="b1">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Olive" title="b2">k2</font></span>)</span>,<font color="Maroon" title="c6">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> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <font color="Maroon" title="c6">s</font>,<font color="Olive" title="b2">k2</font> ) &amp; ( <font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">a</font></span>)</span>,<font color="Olive" title="b1">k1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  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="Olive" title="b3">a</font>,<font color="Olive" title="b1">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Olive" title="b2">k2</font></span>)</span>,<font color="Maroon" title="c6">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> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s</font></span>)</span> ) &amp; <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="Olive" title="b3">a</font>,<font color="Olive" title="b1">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Olive" title="b2">k2</font></span>)</span>,<font color="Maroon" title="c6">s</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b4">b</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">k1</font>, <font color="Maroon" title="c8">k2</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a>  holds <br/> ( ( <font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Maroon" title="c7">k1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  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="Olive" title="b1">a</font>,<font color="Maroon" title="c7">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c8">k2</font></span>)</span>,<font color="Maroon" title="c6">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> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c8">k2</font> ) &amp; ( <font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Maroon" title="c7">k1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  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="Olive" title="b1">a</font>,<font color="Maroon" title="c7">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c8">k2</font></span>)</span>,<font color="Maroon" title="c6">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> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s</font></span>)</span> ) &amp; <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="Olive" title="b1">a</font>,<font color="Maroon" title="c7">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c8">k2</font></span>)</span>,<font color="Maroon" title="c6">s</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">b</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">a</font>, <font color="Maroon" title="c10">b</font> be    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<font color="Maroon" title="c7">k1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  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="c9">a</font>,<font color="Maroon" title="c7">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c8">k2</font></span>)</span>,<font color="Maroon" title="c6">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> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c8">k2</font> ) &amp; ( <font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<font color="Maroon" title="c7">k1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  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="c9">a</font>,<font color="Maroon" title="c7">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c8">k2</font></span>)</span>,<font color="Maroon" title="c6">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> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s</font></span>)</span> ) &amp; <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="c9">a</font>,<font color="Maroon" title="c7">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c8">k2</font></span>)</span>,<font color="Maroon" title="c6">s</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c10">b</font> )</span><br/>









<span class="kw">reconsider </span><font color="Maroon" title="c11">mn</font> = <font color="Maroon" title="c10">b</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_2.html#D2" target="_self" title="SCMPDS_2:def.2">Def2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">S</font> = <font color="Maroon" title="c6">s</font> as   <a href="scmpds_1.html#NM1" title="SCMPDS_1:NM.1">SCMPDS-State</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T155" title="PBOOLE:th.155">PBOOLE:155</a></span>;<br/>
<a NAME="E3:72"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
 ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s</font> &amp;  <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c8">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">k2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_2.html#D20" target="_self" title="SCMPDS_2:def.20">Def20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">da</font> = <font color="Maroon" title="c9">a</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_2.html#D2" target="_self" title="SCMPDS_2:def.2">Def2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">I</font> = <font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c7">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c8">k2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="scmpds_1.html#K1" title="SCMPDS_1:func.1">SCMPDS-Instr</a>  ;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">A2</font> =  <a href="scmpds_1.html#K7" title="SCMPDS_1:func.7">Address_Add</a> <font color="Maroon" title="c12">S</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="scmpds_1.html#K14" title="SCMPDS_1:func.14">P31address</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="scmpds_1.html#K15" title="SCMPDS_1:func.15">P32const</a> </span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">JP</font> =  <a href="scmpds_1.html#K8" title="SCMPDS_1:func.8">jump_address</a> <font color="Maroon" title="c12">S</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="scmpds_1.html#K16" title="SCMPDS_1:func.16">P33const</a> </span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">IF</font> =  <a href="xxreal_0.html#K5" title="XXREAL_0:func.5">IFGT</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">S</font> <a href="scmpds_1.html#K6" title="SCMPDS_1:func.6">.</a> <span class="p2">(<span class="default"><a href="scmpds_1.html#K7" title="SCMPDS_1:func.7">Address_Add</a> <font color="Maroon" title="c12">S</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="scmpds_1.html#K14" title="SCMPDS_1:func.14">P31address</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="scmpds_1.html#K15" title="SCMPDS_1:func.15">P32const</a> </span>)</span></span>)</span></span>)</span>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <span class="p2">(<span class="default"><a href="scmpds_1.html#K3" title="SCMPDS_1:func.3">IC</a> <font color="Maroon" title="c12">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_1.html#K8" title="SCMPDS_1:func.8">jump_address</a> <font color="Maroon" title="c12">S</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="scmpds_1.html#K16" title="SCMPDS_1:func.16">P33const</a> </span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">Da</font> =  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<font color="Maroon" title="c7">k1</font>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">i</font> = 5 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="card_1.html#K7" title="CARD_1:func.7">Segm</a> 14 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/>
<a NAME="E7:72"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
<font color="Maroon" title="c14">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="c19">i</font>,<span class="p2"><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c13">da</font>,<font color="Maroon" title="c7">k1</font>,<font color="Maroon" title="c8">k2</font></span><a href="finseq_1.html#K11" title="FINSEQ_1:func.11">*&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
;<br/>
<a NAME="E8:72"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A3</font></span>: 
( <font color="Maroon" title="c14">I</font> <a href="scmpds_1.html#K14" title="SCMPDS_1:func.14">P31address</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">da</font> &amp; <font color="Maroon" title="c14">I</font> <a href="scmpds_1.html#K15" title="SCMPDS_1:func.15">P32const</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_1.html#T36" title="SCMPDS_1:th.36">SCMPDS_1:36</a></span>;<br/>
<span class="lab"><font color="Green" title="E35">A4</font></span>:  <a href="ami_1.html#K4" title="AMI_1:func.4">Exec</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">a</font>,<font color="Maroon" title="c7">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c8">k2</font></span>)</span>,<font color="Maroon" title="c6">s</font> = 
 <a href="scmpds_1.html#K24" title="SCMPDS_1:func.24">SCM-Exec-Res</a> <font color="Maroon" title="c14">I</font>,<font color="Maroon" title="c12">S</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_1.html#D25" title="SCMPDS_1:def.25">SCMPDS_1:def 25</a></span>
<br/>.= 
 <a href="scmpds_1.html#K4" title="SCMPDS_1:func.4">SCM-Chg</a> <font color="Maroon" title="c12">S</font>,<span class="p1">(<span class="default"><a href="xxreal_0.html#K5" title="XXREAL_0:func.5">IFGT</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">S</font> <a href="scmpds_1.html#K6" title="SCMPDS_1:func.6">.</a> <span class="p3">(<span class="default"><a href="scmpds_1.html#K7" title="SCMPDS_1:func.7">Address_Add</a> <font color="Maroon" title="c12">S</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="scmpds_1.html#K14" title="SCMPDS_1:func.14">P31address</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="scmpds_1.html#K15" title="SCMPDS_1:func.15">P32const</a> </span>)</span></span>)</span></span>)</span>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <span class="p3">(<span class="default"><a href="scmpds_1.html#K3" title="SCMPDS_1:func.3">IC</a> <font color="Maroon" title="c12">S</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="scmpds_1.html#K8" title="SCMPDS_1:func.8">jump_address</a> <font color="Maroon" title="c12">S</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="scmpds_1.html#K16" title="SCMPDS_1:func.16">P33const</a> </span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:72"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="scmpds_1.html#D24" title="SCMPDS_1:def.24">SCMPDS_1:def 24</a></span>
;<br/>
<a NAME="E10:72"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
<font color="Maroon" title="c14">I</font> <a href="scmpds_1.html#K16" title="SCMPDS_1:func.16">P33const</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:72"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="scmpds_1.html#T36" title="SCMPDS_1:th.36">SCMPDS_1:36</a></span>;<br/>
<span class="kw">thus </span><a NAME="E11:72"/>
( <font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<font color="Maroon" title="c7">k1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  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="c9">a</font>,<font color="Maroon" title="c7">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c8">k2</font></span>)</span>,<font color="Maroon" title="c6">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> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c8">k2</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<font color="Maroon" title="c7">k1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  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="c9">a</font>,<font color="Maroon" title="c7">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c8">k2</font></span>)</span>,<font color="Maroon" title="c6">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> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s</font></span>)</span> ) &amp; <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="c9">a</font>,<font color="Maroon" title="c7">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c8">k2</font></span>)</span>,<font color="Maroon" title="c6">s</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c10">b</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:72_2"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<font color="Maroon" title="c7">k1</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="c9">a</font>,<font color="Maroon" title="c7">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c8">k2</font></span>)</span>,<font color="Maroon" title="c6">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> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c8">k2</font></span><br/>

<span class="kw">thus </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="c9">a</font>,<font color="Maroon" title="c7">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c8">k2</font></span>)</span>,<font color="Maroon" title="c6">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> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> = 
 <a href="xxreal_0.html#K5" title="XXREAL_0:func.5">IFGT</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">S</font> <a href="scmpds_1.html#K6" title="SCMPDS_1:func.6">.</a> <span class="p2">(<span class="default"><a href="scmpds_1.html#K7" title="SCMPDS_1:func.7">Address_Add</a> <font color="Maroon" title="c12">S</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="scmpds_1.html#K14" title="SCMPDS_1:func.14">P31address</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="scmpds_1.html#K15" title="SCMPDS_1:func.15">P32const</a> </span>)</span></span>)</span></span>)</span>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <span class="p2">(<span class="default"><a href="scmpds_1.html#K3" title="SCMPDS_1:func.3">IC</a> <font color="Maroon" title="c12">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_1.html#K8" title="SCMPDS_1:func.8">jump_address</a> <font color="Maroon" title="c12">S</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="scmpds_1.html#K16" title="SCMPDS_1:func.16">P33const</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:72"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="scmpds_2.html#T6" target="_self" title="SCMPDS_2:th.6">Th6</a>, <a class="ref" href="scmpds_1.html#T26" title="SCMPDS_1:th.26">SCMPDS_1:26</a></span>
<br/>.= 
 <a href="scmpds_2.html#K18" title="SCMPDS_2:func.18">ICplusConst</a> <font color="Maroon" title="c6">s</font>,<font color="Maroon" title="c8">k2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:72"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E10:72"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E3:72"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E1:72_2"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="ref" href="scmpds_2.html#T6" target="_self" title="SCMPDS_2:th.6">Th6</a>, <a class="ref" href="xxreal_0.html#D11" title="XXREAL_0:def.11">XXREAL_0:def 11</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E12:72"/>
( <font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<font color="Maroon" title="c7">k1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  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="c9">a</font>,<font color="Maroon" title="c7">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c8">k2</font></span>)</span>,<font color="Maroon" title="c6">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> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s</font></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="c9">a</font>,<font color="Maroon" title="c7">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c8">k2</font></span>)</span>,<font color="Maroon" title="c6">s</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c10">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c10">b</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:72_3"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
<font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c9">a</font></span>)</span>,<font color="Maroon" title="c7">k1</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="c9">a</font>,<font color="Maroon" title="c7">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c8">k2</font></span>)</span>,<font color="Maroon" title="c6">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> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s</font></span>)</span></span><br/>

<span class="kw">thus </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="c9">a</font>,<font color="Maroon" title="c7">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c8">k2</font></span>)</span>,<font color="Maroon" title="c6">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> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span> = 
 <a href="xxreal_0.html#K5" title="XXREAL_0:func.5">IFGT</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">S</font> <a href="scmpds_1.html#K6" title="SCMPDS_1:func.6">.</a> <span class="p2">(<span class="default"><a href="scmpds_1.html#K7" title="SCMPDS_1:func.7">Address_Add</a> <font color="Maroon" title="c12">S</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="scmpds_1.html#K14" title="SCMPDS_1:func.14">P31address</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="scmpds_1.html#K15" title="SCMPDS_1:func.15">P32const</a> </span>)</span></span>)</span></span>)</span>,<a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ,<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <span class="p2">(<span class="default"><a href="scmpds_1.html#K3" title="SCMPDS_1:func.3">IC</a> <font color="Maroon" title="c12">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_1.html#K8" title="SCMPDS_1:func.8">jump_address</a> <font color="Maroon" title="c12">S</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">I</font> <a href="scmpds_1.html#K16" title="SCMPDS_1:func.16">P33const</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:72"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="scmpds_2.html#T6" target="_self" title="SCMPDS_2:th.6">Th6</a>, <a class="ref" href="scmpds_1.html#T26" title="SCMPDS_1:th.26">SCMPDS_1:26</a></span>
<br/>.= 
 <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <span class="p1">(<span class="default"><a href="scmpds_1.html#K3" title="SCMPDS_1:func.3">IC</a> <font color="Maroon" title="c12">S</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:72"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E1:72_3"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="ref" href="xxreal_0.html#D11" title="XXREAL_0:def.11">XXREAL_0:def 11</a></span>
<br/>.= 
 <a href="card_1.html#K4" title="CARD_1:func.4">succ</a> <span class="p1">(<span class="default"><a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_2.html#T30" title="AMI_2:th.30">AMI_2:30</a>, <a class="ref" href="funct_7.html#D1" title="FUNCT_7:def.1">FUNCT_7:def 1</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </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="c9">a</font>,<font color="Maroon" title="c7">k1</font> <a href="scmpds_2.html#K9" title="SCMPDS_2:func.9">&lt;=0_goto</a> <font color="Maroon" title="c8">k2</font></span>)</span>,<font color="Maroon" title="c6">s</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c10">b</font> = 
<font color="Maroon" title="c12">S</font> <a href="scmpds_1.html#K6" title="SCMPDS_1:func.6">.</a> <font color="Maroon" title="c11">mn</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:72"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="scmpds_1.html#T27" title="SCMPDS_1:th.27">SCMPDS_1:27</a></span>
<br/>.= 
<font color="Maroon" title="c6">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c10">b</font>

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
