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<span class="kw">let </span><font color="Maroon" title="c3">P</font> be   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">s</font> being   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="memstr_0.html#V5" title="MEMSTR_0:attr.5">-started</a>  <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a><br/>  for <font color="Olive" title="b2">I</font> being   <a href="compos_1.html#NM4" title="COMPOS_1:NM.4">Program</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a><br/>  for <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a><br/>  for <font color="Olive" title="b5">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">a</font></span>)</span>,<font color="Olive" title="b5">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Olive" title="b3">a</font>,<font color="Olive" title="b5">i</font>,<font color="Olive" title="b2">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<font color="Olive" title="b1">s</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">s</font> be   <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="memstr_0.html#V5" title="MEMSTR_0:attr.5">-started</a>  <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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#NM4" title="COMPOS_1:NM.4">Program</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a><br/>  for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a><br/>  for <font color="Olive" title="b4">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">a</font></span>)</span>,<font color="Olive" title="b4">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">i</font>,<font color="Olive" title="b1">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c5">I</font> be   <a href="compos_1.html#NM4" title="COMPOS_1:NM.4">Program</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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>, <font color="Olive" title="b2">b</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a><br/>  for <font color="Olive" title="b3">i</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Olive" title="b3">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font> be   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Olive" title="b1">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c8">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font> )</span><br/>









<span class="kw">assume </span><a NAME="E1:34"/>
<font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font></span>)</span>,<font color="Maroon" title="c8">i</font>)</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font></span><br/>

<a NAME="E2:34"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<span class="p1">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">s</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K7" title="MEMSTR_0:func.7">Start-At</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_8.html#T21" target="_self" onmouseover="rs('scmpds_8/T21')" onmouseout="rh()">Th19</a></span>;<br/>
<a NAME="E3:34"/>
 not <font color="Maroon" title="c7">b</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K7" title="MEMSTR_0:func.7">Start-At</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">I</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>,<a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_4.html#T18" onmouseover="rs('scmpds_4/T18')" onmouseout="rh()">SCMPDS_4:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:34"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<span class="p2">(<span class="default"><a href="scmpds_8.html#K2" title="SCMPDS_8:func.2">while&gt;0</a> (<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">s</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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