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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#V2" title="COMPOS_1:attr.2">halt-free</a>   <a href="scmpds_4.html#V3" title="SCMPDS_4:attr.3">shiftable</a>  <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> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a><br/>  for <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">c</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/>  for <font color="Olive" title="b6">X</font>, <font color="Olive" title="b7">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st (  for <font color="Olive" title="b8">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b6">X</font> holds <br/><font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b5">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="b4">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b8">t</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="b9">Q</font> being   <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>  st (  for <font color="Olive" title="b10">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b10">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b6">X</font> holds <br/><font color="Olive" title="b8">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b10">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b5">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b8">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="b4">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b10">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b10">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b7">Y</font> holds <br/><font color="Olive" title="b8">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b10">x</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="b10">x</font> ) &amp; <font color="Olive" title="b8">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">a</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="b3">a</font> &amp; <font color="Olive" title="b8">t</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="b4">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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> (<font color="Olive" title="b2">I</font>,<font color="Olive" title="b9">Q</font>,<font color="Olive" title="b8">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">a</font> &amp; <font color="Olive" title="b2">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b8">t</font>,<font color="Olive" title="b9">Q</font> &amp; <font color="Olive" title="b2">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b8">t</font>,<font color="Olive" title="b9">Q</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Olive" title="b2">I</font>,<font color="Olive" title="b9">Q</font>,<font color="Olive" title="b8">t</font>)</span>)</span> <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="b4">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b8">t</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="b4">i</font>)</span>)</span> &amp; (  for <font color="Olive" title="b10">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b10">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b6">X</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Olive" title="b2">I</font>,<font color="Olive" title="b9">Q</font>,<font color="Olive" title="b8">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b10">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b5">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Olive" title="b2">I</font>,<font color="Olive" title="b9">Q</font>,<font color="Olive" title="b8">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="b4">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b10">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b10">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b7">Y</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Olive" title="b2">I</font>,<font color="Olive" title="b9">Q</font>,<font color="Olive" title="b8">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b8">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b10">x</font> ) ) ) holds <br/>(  <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="b4">i</font>,<font color="Olive" title="b2">I</font>) <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c3">P</font> &amp;  <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="b4">i</font>,<font color="Olive" title="b2">I</font>) <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c3">P</font> &amp; ( <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="b4">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <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="Olive" title="b3">a</font>,<font color="Olive" title="b4">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>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="Olive" title="b3">a</font>,<font color="Olive" title="b4">i</font>,<font color="Olive" title="b2">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Olive" title="b2">I</font>,<font color="Maroon" title="c3">P</font>,<font color="Olive" title="b1">s</font>)</span>)</span></span>)</span>) ) )</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#V2" title="COMPOS_1:attr.2">halt-free</a>   <a href="scmpds_4.html#V3" title="SCMPDS_4:attr.3">shiftable</a>  <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> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a><br/>  for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">c</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/>  for <font color="Olive" title="b5">X</font>, <font color="Olive" title="b6">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st (  for <font color="Olive" title="b7">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b5">X</font> holds <br/><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b4">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="b3">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b7">t</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="b8">Q</font> being   <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>  st (  for <font color="Olive" title="b9">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b5">X</font> holds <br/><font color="Olive" title="b7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b4">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="b3">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b9">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b6">Y</font> holds <br/><font color="Olive" title="b7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">x</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="b9">x</font> ) &amp; <font color="Olive" title="b7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">a</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">a</font> &amp; <font color="Olive" title="b7">t</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="b3">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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> (<font color="Olive" title="b1">I</font>,<font color="Olive" title="b8">Q</font>,<font color="Olive" title="b7">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">a</font> &amp; <font color="Olive" title="b1">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b7">t</font>,<font color="Olive" title="b8">Q</font> &amp; <font color="Olive" title="b1">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b7">t</font>,<font color="Olive" title="b8">Q</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Olive" title="b1">I</font>,<font color="Olive" title="b8">Q</font>,<font color="Olive" title="b7">t</font>)</span>)</span> <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="b3">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b7">t</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="b3">i</font>)</span>)</span> &amp; (  for <font color="Olive" title="b9">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b5">X</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Olive" title="b1">I</font>,<font color="Olive" title="b8">Q</font>,<font color="Olive" title="b7">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b4">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Olive" title="b1">I</font>,<font color="Olive" title="b8">Q</font>,<font color="Olive" title="b7">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="b3">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b9">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b9">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b6">Y</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Olive" title="b1">I</font>,<font color="Olive" title="b8">Q</font>,<font color="Olive" title="b7">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b9">x</font> ) ) ) holds <br/>(  <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="b3">i</font>,<font color="Olive" title="b1">I</font>) <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp;  <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="b3">i</font>,<font color="Olive" title="b1">I</font>) <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp; ( <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="b3">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <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="Olive" title="b2">a</font>,<font color="Olive" title="b3">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>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="Olive" title="b2">a</font>,<font color="Olive" title="b3">i</font>,<font color="Olive" title="b1">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Olive" title="b1">I</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">s</font>)</span>)</span></span>)</span>) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">I</font> be   <a href="compos_1.html#V2" title="COMPOS_1:attr.2">halt-free</a>   <a href="scmpds_4.html#V3" title="SCMPDS_4:attr.3">shiftable</a>  <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> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a><br/>  for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">c</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/>  for <font color="Olive" title="b4">X</font>, <font color="Olive" title="b5">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st (  for <font color="Olive" title="b6">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">X</font> holds <br/><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="b2">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b6">t</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="b7">Q</font> being   <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>  st (  for <font color="Olive" title="b8">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">X</font> holds <br/><font color="Olive" title="b6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="b2">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b8">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b5">Y</font> holds <br/><font color="Olive" title="b6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">x</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="b8">x</font> ) &amp; <font color="Olive" title="b6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</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="b1">a</font> &amp; <font color="Olive" title="b6">t</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="b2">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b7">Q</font>,<font color="Olive" title="b6">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b6">t</font>,<font color="Olive" title="b7">Q</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b6">t</font>,<font color="Olive" title="b7">Q</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b7">Q</font>,<font color="Olive" title="b6">t</font>)</span>)</span> <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="b2">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b6">t</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="b2">i</font>)</span>)</span> &amp; (  for <font color="Olive" title="b8">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">X</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b7">Q</font>,<font color="Olive" title="b6">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b7">Q</font>,<font color="Olive" title="b6">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="b2">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b8">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b8">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b5">Y</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b7">Q</font>,<font color="Olive" title="b6">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b8">x</font> ) ) ) holds <br/>(  <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="b2">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp;  <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="b2">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp; ( <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="b2">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <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="Olive" title="b1">a</font>,<font color="Olive" title="b2">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>  <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="Olive" title="b1">a</font>,<font color="Olive" title="b2">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">s</font>)</span>)</span></span>)</span>) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">a</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>, <font color="Olive" title="b2">c</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><br/>  for <font color="Olive" title="b3">X</font>, <font color="Olive" title="b4">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st (  for <font color="Olive" title="b5">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">X</font> holds <br/><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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></span>)</span> ) &amp; (  for <font color="Olive" title="b5">t</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="b6">Q</font> being   <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>  st (  for <font color="Olive" title="b7">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">X</font> holds <br/><font color="Olive" title="b5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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></span>)</span> ) &amp; (  for <font color="Olive" title="b7">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">Y</font> holds <br/><font color="Olive" title="b5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">x</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="b7">x</font> ) &amp; <font color="Olive" title="b5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</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="c6">a</font> &amp; <font color="Olive" title="b5">t</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#NR4" title="XXREAL_0:NR.4">&gt;</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> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b6">Q</font>,<font color="Olive" title="b5">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b5">t</font>,<font color="Olive" title="b6">Q</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b5">t</font>,<font color="Olive" title="b6">Q</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b6">Q</font>,<font color="Olive" title="b5">t</font>)</span>)</span> <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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">t</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> &amp; (  for <font color="Olive" title="b7">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">X</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b6">Q</font>,<font color="Olive" title="b5">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b6">Q</font>,<font color="Olive" title="b5">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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></span>)</span> ) &amp; (  for <font color="Olive" title="b7">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">Y</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b6">Q</font>,<font color="Olive" title="b5">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">x</font> ) ) ) holds <br/>(  <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>) <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp;  <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>) <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp; ( <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#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <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="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>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="Olive" title="b1">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">s</font>)</span>)</span></span>)</span>) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c8">c</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">  for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st (  for <font color="Olive" title="b3">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">X</font> holds <br/><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="c7">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b3">t</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="b4">Q</font> being   <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>  st (  for <font color="Olive" title="b5">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">X</font> holds <br/><font color="Olive" title="b3">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="c7">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b5">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">Y</font> holds <br/><font color="Olive" title="b3">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">x</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="b5">x</font> ) &amp; <font color="Olive" title="b3">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</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="c6">a</font> &amp; <font color="Olive" title="b3">t</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="c7">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b4">Q</font>,<font color="Olive" title="b3">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b3">t</font>,<font color="Olive" title="b4">Q</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b3">t</font>,<font color="Olive" title="b4">Q</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b4">Q</font>,<font color="Olive" title="b3">t</font>)</span>)</span> <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="c7">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">t</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="c7">i</font>)</span>)</span> &amp; (  for <font color="Olive" title="b5">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">X</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b4">Q</font>,<font color="Olive" title="b3">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b4">Q</font>,<font color="Olive" title="b3">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="c7">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b5">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b5">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">Y</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b4">Q</font>,<font color="Olive" title="b3">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">x</font> ) ) ) holds <br/>(  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp;  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp; ( <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="c7">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <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="c7">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>  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">s</font>)</span>)</span></span>)</span>) ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">X</font>, <font color="Maroon" title="c10">Y</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">X</font> holds <br/><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="c7">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b1">t</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">Q</font> being   <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>  st (  for <font color="Olive" title="b3">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">X</font> holds <br/><font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="c7">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b3">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">Y</font> holds <br/><font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</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">x</font> ) &amp; <font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</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="c6">a</font> &amp; <font color="Olive" title="b1">t</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="c7">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b1">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b1">t</font>,<font color="Olive" title="b2">Q</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b1">t</font>,<font color="Olive" title="b2">Q</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b1">t</font>)</span>)</span> <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="c7">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">t</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="c7">i</font>)</span>)</span> &amp; (  for <font color="Olive" title="b3">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">X</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b1">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b1">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="c7">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b3">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">Y</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b1">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> ) ) ) implies (  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp;  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp; ( <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="c7">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <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="c7">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>  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">s</font>)</span>)</span></span>)</span>) ) ) )</span><br/>











<span class="kw">set </span><font color="Maroon" title="c11">b</font> =  <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p1">(<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="c7">i</font>);<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <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>] <span class="kw">means </span>( (  for <font color="Olive" title="b1">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">X</font> holds <br/>$1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default">$1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="c7">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b1">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">Y</font> holds <br/>$1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</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="b1">x</font> ) );<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p2">(<span class="default"><a href="memstr_0.html#K2" title="MEMSTR_0:func.2">the_Values_of</a> <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a></span>)</span></span>)</span>,<a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a><span class="kw"> such that </span><br/><a NAME="E2:41"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
 for <font color="Olive" title="b1">s</font> being   <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> holds <br/> ( ( <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="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="c7">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 <font color="Maroon" title="c12">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  ) &amp; ( <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="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="c7">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies <font color="Maroon" title="c12">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Olive" title="b1">s</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> <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="c7">i</font>)</span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_8.html#T5" target="_self" onmouseover="rs('scmpds_8/T5')" onmouseout="rh()">Th3</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <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>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  = <font color="Maroon" title="c12">f</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> $1;<br/>
<a NAME="E3:41"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
 for <font color="Olive" title="b1">t</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>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>] &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/><font color="Olive" title="b1">t</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="c7">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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E21">A1</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:41"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">X</font> holds <br/><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="c7">i</font>)</span>)</span></span>)</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"> (  ex <font color="Olive" title="b1">t</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> ex <font color="Olive" title="b2">Q</font> being   <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> st <br/>( (  for <font color="Olive" title="b3">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">X</font> holds <br/><font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="c7">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b3">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">Y</font> holds <br/><font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</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">x</font> ) &amp; <font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</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="c6">a</font> &amp; <font color="Olive" title="b1">t</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="c7">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp;  not <!--HUMANRECFAILED-->( <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b1">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b1">t</font>,<font color="Olive" title="b2">Q</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b1">t</font>,<font color="Olive" title="b2">Q</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b1">t</font>)</span>)</span> <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="c7">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">t</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="c7">i</font>)</span>)</span> &amp; (  for <font color="Olive" title="b3">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">X</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b1">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b1">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="c7">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b3">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">Y</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b1">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> ) ) ) or (  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp;  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp; ( <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="c7">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <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="c7">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>  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">s</font>)</span>)</span></span>)</span>) ) ) )</span><br/>

<a NAME="E5:41"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">s</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E23">A3</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:41"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
 for <font color="Olive" title="b1">t</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">Q</font> being   <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>  st (  for <font color="Olive" title="b3">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">X</font> holds <br/><font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="c7">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b3">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">Y</font> holds <br/><font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</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">x</font> ) &amp; <font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</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="c6">a</font> &amp; <font color="Olive" title="b1">t</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="c7">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b1">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b1">t</font>,<font color="Olive" title="b2">Q</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b1">t</font>,<font color="Olive" title="b2">Q</font> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b1">t</font>)</span>)</span> <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="c7">i</font>)</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">t</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="c7">i</font>)</span>)</span> &amp; (  for <font color="Olive" title="b3">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">X</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b1">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b1">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="c7">i</font>)</span>)</span></span>)</span> ) &amp; (  for <font color="Olive" title="b3">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">Y</font> holds <br/><span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b1">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">x</font> ) )
 ; <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"> (  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp;  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp; ( <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="c7">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <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="c7">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>  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">s</font>)</span>)</span></span>)</span>) ) )</span><br/>

<div><span class="lab"><font color="Green" title="E26">A6</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_1"><span class="kw">now </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">  for <font color="Olive" title="b1">t</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">Q</font> being   <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>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>] &amp; <font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</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="c6">a</font> &amp; <font color="Olive" title="b1">t</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="c7">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b1">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b1">t</font>,<font color="Olive" title="b2">Q</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b1">t</font>,<font color="Olive" title="b2">Q</font> &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b1">t</font>)</span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">t</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b2">Q</font>,<font color="Olive" title="b1">t</font>)</span>)</span>] )</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c13">t</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">Q</font> being   <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>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">t</font>] &amp; <font color="Maroon" title="c13">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</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="c6">a</font> &amp; <font color="Maroon" title="c13">t</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="c7">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b1">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c13">t</font>,<font color="Olive" title="b1">Q</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c13">t</font>,<font color="Olive" title="b1">Q</font> &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b1">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">t</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Olive" title="b1">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span>] )</span><br/><span class="kw">let </span><font color="Maroon" title="c14">Q</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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">t</font>] &amp; <font color="Maroon" title="c13">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</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="c6">a</font> &amp; <font color="Maroon" title="c13">t</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="c7">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c13">t</font>,<font color="Maroon" title="c14">Q</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c13">t</font>,<font color="Maroon" title="c14">Q</font> &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">t</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span>] ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:41_1"/><span class="lab"><font color="Green" title="E27">A7</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">t</font>]
 <span class="kw">and </span><br/><a NAME="E2:41_1"/><span class="lab"><font color="Green" title="E28">A8</font></span>: 
<font color="Maroon" title="c13">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</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="c6">a</font>
 <span class="kw">and </span><br/><a NAME="E3:41_1"/><span class="lab"><font color="Green" title="E29">A9</font></span>: 
<font color="Maroon" title="c13">t</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="c7">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c13">t</font>,<font color="Maroon" title="c14">Q</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c13">t</font>,<font color="Maroon" title="c14">Q</font> &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">t</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span>] )</span><br/><span class="kw">set </span><font color="Maroon" title="c15">It</font> =  <a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>);<br/><span class="kw">set </span><font color="Maroon" title="c16">t2</font> =  <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c17">t1</font> = <font color="Maroon" title="c13">t</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c18">v</font> being   <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><span class="kw"> such that </span><br/><a NAME="E5:41_1"/><span class="lab"><font color="Green" title="E30">A10</font></span>: 
<font color="Maroon" title="c18">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">t</font>
 <span class="kw">and </span><br/><a NAME="E6:41_1"/><span class="lab"><font color="Green" title="E31">A11</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">X</font> holds <br/><font color="Maroon" title="c18">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="c7">i</font>)</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:41_1"/><span class="lab"><font color="Green" title="E32">A12</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">Y</font> holds <br/><font color="Maroon" title="c18">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</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="b1">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E8:41_1"/>
( <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c13">t</font>,<font color="Maroon" title="c14">Q</font> &amp; <font color="Maroon" title="c5">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c13">t</font>,<font color="Maroon" title="c14">Q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E2:41_1"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E5:41_1"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="txt" href="#E7:41_1"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="txt" href="#E6:41_1"><span class="lab"><font color="Green" title="E31">A11</font></span></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"> ( <font color="Maroon">H<sub>1</sub></font>( <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">t</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span>] )</span><br/><span class="kw">thus </span><a NAME="E9:41_1"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span>) <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">t</font>)
  <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">S<sub>1</sub></font>[ <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span>]</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:41_1_1"/><span class="lab"><font color="Green" title="E33">A13</font></span>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">t</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="c7">i</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:41_1_1"/><span class="lab"><font color="Green" title="E34">A14</font></span>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c13">t</font>)
 ; <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"> contradiction</span><br/>

<font color="Maroon">H<sub>1</sub></font>( <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span>) = 
<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span></span>)</span> <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="c7">i</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E1:41_1_1"><span class="lab"><font color="Green" title="E33">A13</font></span></a>, <a class="txt" href="#E2:41_1_1"><span class="lab"><font color="Green" title="E34">A14</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span> <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="c7">i</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_5.html#T15" onmouseover="rs('scmpds_5/T15')" onmouseout="rh()">SCMPDS_5:15</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:41_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E2:41_1"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E1:41_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="txt" href="#E2:41_1_1"><span class="lab"><font color="Green" title="E34">A14</font></span></a>, <a class="txt" href="#E1:41_1_1"><span class="lab"><font color="Green" title="E33">A13</font></span></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/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E10:41_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</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><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c19">v</font> =  <a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span>;<br/>
<div><span class="kw">hereby </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">  for <font color="Olive" title="b1">x</font> being   <a href="scmpds_2.html#NM1" title="SCMPDS_2:NM.1">Int_position</a>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">Y</font> holds <br/><span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</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="b1">x</font></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c20">x</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"> ( <font color="Maroon" title="c20">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">X</font> implies <span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="c7">i</font>)</span>)</span></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:41_1_2_1"/>
<font color="Maroon" title="c20">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">X</font>
 ; <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="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="c7">i</font>)</span>)</span></span>)</span></span><br/><a NAME="E2:41_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="c7">i</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E2:41_1"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E1:41_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a></span>;<br/><a NAME="E3:41_1_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="c7">i</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_5.html#T15" onmouseover="rs('scmpds_5/T15')" onmouseout="rh()">SCMPDS_5:15</a></span>;<br/><span class="kw">hence </span><a NAME="E4:41_1_2_1"/>
<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">c</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p3">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">DataLoc</a> (<span class="p3">(<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="c7">i</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_5.html#T15" onmouseover="rs('scmpds_5/T15')" onmouseout="rh()">SCMPDS_5:15</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/></div>
<span class="kw">end;</span></div>

<div><span class="kw">hereby </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>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c20">x</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"> ( <font color="Maroon" title="c20">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">Y</font> implies <span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</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="c20">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:41_1_2_2"/><span class="lab"><font color="Green" title="E33">A15</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c10">Y</font>
 ; <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="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</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="c20">x</font></span><br/><a NAME="E2:41_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E2:41_1"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a>, <a class="txt" href="#E1:41_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a></span>;<br/><a NAME="E3:41_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_5.html#T15" onmouseover="rs('scmpds_5/T15')" onmouseout="rh()">SCMPDS_5:15</a></span>;<br/><span class="kw">hence </span><a NAME="E4:41_1_2_2"/>
<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c14">Q</font>,<font color="Maroon" title="c13">t</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</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="c20">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_1"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="txt" href="#E7:41_1"><span class="lab"><font color="Green" title="E32">A12</font></span></a>, <a class="txt" href="#E1:41_1_2_2"><span class="lab"><font color="Green" title="E33">A15</font></span></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/></div>
<span class="kw">end;</span></div>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E8:41"/>
(  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp;  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> )
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="scmpds_8.html#S3" target="_self" onmouseover="rs('scmpds_8/S3')" onmouseout="rh()">SCMPDS_8:sch 3</a>(<a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E5:41"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E7:41"><span class="lab"><font color="Green" title="E26">A6</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E9:41"/>
(  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> &amp;  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>) <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c4">s</font>,<font color="Maroon" title="c3">P</font> )
 ; <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="c7">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  implies  <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="c7">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>  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">s</font>)</span>)</span></span>)</span>) )</span><br/>

<span class="kw">assume </span><a NAME="E10:41"/><span class="lab"><font color="Green" title="E27">A16</font></span>: 
<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="c7">i</font>)</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</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">  <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="c7">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>  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">s</font>)</span>)</span></span>)</span>)</span><br/>

<a NAME="E11:41"/>
 <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="c7">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>  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">s</font>)</span>)</span></span>)</span>)
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="scmpds_8.html#S4" target="_self" onmouseover="rs('scmpds_8/S4')" onmouseout="rh()">SCMPDS_8:sch 4</a>(<a class="txt" href="#E10:41"><span class="lab"><font color="Green" title="E27">A16</font></span></a>, <a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E5:41"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E7:41"><span class="lab"><font color="Green" title="E26">A6</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E12:41"/>
 <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="c7">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>  <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="c7">i</font>,<font color="Maroon" title="c5">I</font>)</span>)</span>,<font color="Maroon" title="c3">P</font>,<span class="p1">(<span class="default"><a href="memstr_0.html#K8" title="MEMSTR_0:func.8">Initialize</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K6" title="SCMPDS_4:func.6">IExec</a> (<font color="Maroon" title="c5">I</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">s</font>)</span>)</span></span>)</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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