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<span class="kw">let </span><font color="Maroon" title="c17">s</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">I</font> being  <a href="ami_1.html#V6" title="AMI_1:attr.6">halt-free</a>   <a href="scmpds_4.html#V3" title="SCMPDS_4:attr.3">shiftable</a>  <a href="scmnorm.html#NM2" title="SCMNORM:NM.2">Program</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font> being   <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a> <br/> for <font color="Olive" title="b5">i</font>, <font color="Olive" title="b6">d</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">I</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">d</font> &amp; <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b4">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Olive" title="b6">d</font>,<font color="Olive" title="b5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b4">c</font></span>)</span> &amp; ( for <font color="Olive" title="b7">t</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>   st <font color="Olive" title="b7">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b7">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b4">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b7">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">d</font> &amp; <font color="Olive" title="b7">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Olive" title="b6">d</font>,<font color="Olive" title="b5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b4">c</font></span>)</span> &amp; <font color="Olive" title="b7">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b7">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b4">c</font> holds <br/>( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Olive" title="b1">I</font>,<font color="Olive" title="b7">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">d</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> &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> &amp; ( <font color="Olive" title="b7">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b7">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b4">c</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Olive" title="b1">I</font>,<font color="Olive" title="b7">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b4">c</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Olive" title="b1">I</font>,<font color="Olive" title="b7">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b4">c</font> ) ) &amp; ( <font color="Olive" title="b7">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b7">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b4">c</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Olive" title="b1">I</font>,<font color="Olive" title="b7">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b4">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b7">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">b</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Olive" title="b1">I</font>,<font color="Olive" title="b7">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">b</font> ) ) &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Olive" title="b1">I</font>,<font color="Olive" title="b7">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Olive" title="b6">d</font>,<font color="Olive" title="b5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Olive" title="b1">I</font>,<font color="Olive" title="b7">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Olive" title="b1">I</font>,<font color="Olive" title="b7">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b4">c</font></span>)</span> ) ) holds <br/>(  <a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b5">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="c17">s</font> &amp;  <a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b5">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="c17">s</font> &amp; ( <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">a</font></span>)</span>,<font color="Olive" title="b5">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b5">i</font>,<font color="Olive" title="b1">I</font></span>)</span>,<font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b5">i</font>,<font color="Olive" title="b1">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Olive" title="b1">I</font>,<font color="Maroon" title="c17">s</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c18">I</font> be   <a href="ami_1.html#V6" title="AMI_1:attr.6">halt-free</a>   <a href="scmpds_4.html#V3" title="SCMPDS_4:attr.3">shiftable</a>  <a href="scmnorm.html#NM2" title="SCMNORM:NM.2">Program</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a> <br/> for <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">d</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">I</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">d</font> &amp; <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Olive" title="b5">d</font>,<font color="Olive" title="b4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">c</font></span>)</span> &amp; ( for <font color="Olive" title="b6">t</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>   st <font color="Olive" title="b6">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b6">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b6">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">d</font> &amp; <font color="Olive" title="b6">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Olive" title="b5">d</font>,<font color="Olive" title="b4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">c</font></span>)</span> &amp; <font color="Olive" title="b6">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b6">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">c</font> holds <br/>( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b6">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">d</font> &amp; <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b6">t</font> &amp; <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b6">t</font> &amp; ( <font color="Olive" title="b6">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b6">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">c</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b6">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">c</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b6">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">c</font> ) ) &amp; ( <font color="Olive" title="b6">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b6">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">c</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b6">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">b</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b6">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">b</font> ) ) &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b6">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Olive" title="b5">d</font>,<font color="Olive" title="b4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b6">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b6">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b3">c</font></span>)</span> ) ) holds <br/>(  <a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">i</font>,<font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c17">s</font> &amp;  <a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">i</font>,<font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c17">s</font> &amp; ( <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Olive" title="b4">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c17">s</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c19">a</font>, <font color="Maroon" title="c20">b</font>, <font color="Maroon" title="c21">c</font> be    <a href="scmpds_2.html#M1" title="SCMPDS_2:mode.1">Int_position</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">d</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">I</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">d</font> &amp; <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Olive" title="b2">d</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> &amp; ( for <font color="Olive" title="b3">t</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>   st <font color="Olive" title="b3">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b3">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b3">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">d</font> &amp; <font color="Olive" title="b3">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Olive" title="b2">d</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> &amp; <font color="Olive" title="b3">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> holds <br/>( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b3">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">d</font> &amp; <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b3">t</font> &amp; <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b3">t</font> &amp; ( <font color="Olive" title="b3">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b3">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b3">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b3">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> ) ) &amp; ( <font color="Olive" title="b3">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b3">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b3">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> ) ) &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b3">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Olive" title="b2">d</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b3">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b3">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> ) ) holds <br/>(  <a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c17">s</font> &amp;  <a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c17">s</font> &amp; ( <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font></span>)</span>,<font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c17">s</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c22">i</font>, <font color="Maroon" title="c23">d</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">I</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">d</font> &amp; <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>   st <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">d</font> &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> holds <br/>( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">d</font> &amp; <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b1">t</font> &amp; <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b1">t</font> &amp; ( <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> ) ) &amp; ( <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> ) ) &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> ) ) implies (  <a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c17">s</font> &amp;  <a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c17">s</font> &amp; ( <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font></span>)</span>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c17">s</font></span>)</span> ) ) )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c24">ci</font> =  <a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font></span>)</span>,<font color="Maroon" title="c22">i</font>;<br/>
<span class="kw">assume </span><a NAME="E1:42"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c18">I</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">d</font> or  not <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  not <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  not <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> or  ex <font color="Olive" title="b1">t</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  st <br/>( <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">d</font> &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> &amp; not <!--HUMANRECFAILED-->( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">d</font> &amp; <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b1">t</font> &amp; <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b1">t</font> &amp; ( <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> ) ) &amp; ( <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> ) ) &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> ) ) or (  <a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c17">s</font> &amp;  <a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c17">s</font> &amp; ( <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font></span>)</span>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c17">s</font></span>)</span> ) ) )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c25">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> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> </span>)</span>,<a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:42"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
for <font color="Olive" title="b1">s</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  holds <br/> ( ( <font color="Olive" title="b1">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> implies <font color="Maroon" title="c25">f</font> <a href="scpinvar.html#K2" title="SCPINVAR:func.2">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; ( <font color="Olive" title="b1">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> implies <font color="Maroon" title="c25">f</font> <a href="scpinvar.html#K2" title="SCPINVAR:func.2">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scpinvar.html#T6" target="_self" title="SCPINVAR:th.6">Th6</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  = <font color="Maroon" title="c25">f</font> <a href="scpinvar.html#K2" title="SCPINVAR:func.2">.</a> $1;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span> ex <font color="Olive" title="b1">t</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  st <br/>( <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> );<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:42"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">d</font>
 <span class="kw">and </span><br/><a NAME="E5:42"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E6:42"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E7:42"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
<font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">t</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>  st <br/>( <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">d</font> &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> &amp; not <!--HUMANRECFAILED-->( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">d</font> &amp; <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b1">t</font> &amp; <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b1">t</font> &amp; ( <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> ) ) &amp; ( <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> ) ) &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> ) ) or (  <a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c17">s</font> &amp;  <a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c17">s</font> &amp; ( <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font></span>)</span>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c17">s</font></span>)</span> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E8:42"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
for <font color="Olive" title="b1">t</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>   st <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">d</font> &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> &amp; <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> holds <br/>( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">d</font> &amp; <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Olive" title="b1">t</font> &amp; <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Olive" title="b1">t</font> &amp; ( <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> ) ) &amp; ( <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> ) ) &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c17">s</font> &amp;  <a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c17">s</font> &amp; ( <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font></span>)</span>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c17">s</font></span>)</span> ) )</span><br/>

<div><span class="lab"><font color="Green" title="E32">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c26">t</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>] &amp; <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> &amp; <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font></span>)</span>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> &amp; <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c26">t</font> &amp; <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c26">t</font> &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">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>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span>] ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42_1"/><span class="lab"><font color="Green" title="E33">A9</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>]
 <span class="kw">and </span><br/><a NAME="E2:42_1"/><span class="lab"><font color="Green" title="E34">A10</font></span>: 
<font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font>
 <span class="kw">and </span><br/><a NAME="E3:42_1"/><span class="lab"><font color="Green" title="E35">A11</font></span>: 
<font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font></span>)</span>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> &amp; <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c26">t</font> &amp; <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c26">t</font> &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">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>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span>] )</span><br/><span class="kw">set </span><font color="Maroon" title="c27">It</font> =  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font>;<br/><span class="kw">set </span><font color="Maroon" title="c28">t2</font> =  <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c29">t1</font> =  <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>;<br/><span class="kw">set </span><font color="Maroon" title="c30">x</font> = <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font>;<br/><span class="kw">set </span><font color="Maroon" title="c31">y</font> = <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font>;<br/><span class="kw">consider </span><font color="Maroon" title="c32">v</font> being   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <span class="kw"> such that </span><br/><a NAME="E5:42_1"/><span class="lab"><font color="Green" title="E36">A12</font></span>: 
<font color="Maroon" title="c32">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>
 <span class="kw">and </span><br/><a NAME="E6:42_1"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
<font color="Maroon" title="c32">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E7:42_1"/><span class="lab"><font color="Green" title="E38">A14</font></span>: 
<font color="Maroon" title="c32">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E8:42_1"/><span class="lab"><font color="Green" title="E39">A15</font></span>: 
<font color="Maroon" title="c32">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1"><span class="lab"><font color="Green" title="E33">A9</font></span></a></span>;<br/><a NAME="E9:42_1"/><span class="lab"><font color="Green" title="E40">A16</font></span>: 
<font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E6:42_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>;<br/><a NAME="E10:42_1"/><span class="lab"><font color="Green" title="E41">A17</font></span>: 
<font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E7:42_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>;<br/><span class="lab"><font color="Green" title="E42">A18</font></span>: <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c32">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E8:42_1"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c32">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>
;<br/><a NAME="E12:42_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A19</font></span>: 
<font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E3:42_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E13:42_1"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E9:42_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E10:42_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E11:42_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c26">t</font> &amp; <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c26">t</font> &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">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>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span>] )</span><br/><span class="kw">thus </span><a NAME="E14:42_1"/>
( <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c26">t</font> &amp; <font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c26">t</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E9:42_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E10:42_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E11:42_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E12:42_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">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>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>) &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span>] )</span><br/><div><span class="lab"><font color="Green" title="E44">A20</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_1_2_1"/>
( <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> or <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_2_1_1"/><span class="lab"><font color="Green" title="E45">A21</font></span>: 
<font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> )</span><br/><div class="add"><a NAME="E2:42_1_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></span>;<br/><span class="kw">hence </span><a NAME="E3:42_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E9:42_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E10:42_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E11:42_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E1:42_1_2_1_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> )</span><br/><span class="kw">thus </span><a NAME="E4:42_1_2_1_1"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E9:42_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E10:42_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E11:42_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E1:42_1_2_1_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span><br/><a NAME="E5:42_1_2_1_1"/><span class="lab"><font color="Green" title="E46">A22</font></span>: 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E9:42_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E10:42_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E11:42_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E1:42_1_2_1_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><a NAME="E1:42_1_2_1_1_1"/><span class="lab"><font color="Green" title="E47">A23</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_2_1_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="ref" href="xxreal_0.html#D10" title="XXREAL_0:def.10">XXREAL_0:def 10</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_1_2_1_1_1_1"/>
(  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> or  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_2_1_1_1_1_1"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_1_2_1_1_1_1_1"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E5:42_1_2_1_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E1:42_1_2_1_1_1"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_2_1_1_1_1_2"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_1_2_1_1_1_1_2"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E9:42_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E10:42_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E11:42_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E1:42_1_2_1_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E1:42_1_2_1_1_1"><span class="lab"><font color="Green" title="E47">A23</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_2_1_2"/><span class="lab"><font color="Green" title="E45">A24</font></span>: 
<font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E9:42_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E10:42_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E11:42_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E12:42_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> )</span><br/><a NAME="E3:42_1_2_1_2"/>
<font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:42_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E1:42_1_2_1_2"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:42_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T52" title="XREAL_1:th.52">XREAL_1:52</a></span>;<br/><span class="kw">hence </span><a NAME="E5:42_1_2_1_2"/>
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E9:42_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E10:42_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E11:42_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E12:42_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E1:42_1_2_1_2"><span class="lab"><font color="Green" title="E45">A24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span><br/><a NAME="E6:42_1_2_1_2"/><span class="lab"><font color="Green" title="E46">A25</font></span>: 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E9:42_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E10:42_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E11:42_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E12:42_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E1:42_1_2_1_2"><span class="lab"><font color="Green" title="E45">A24</font></span></a></span>;<br/><a NAME="E7:42_1_2_1_2"/><span class="lab"><font color="Green" title="E47">A26</font></span>: 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E9:42_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E10:42_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E11:42_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E12:42_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E1:42_1_2_1_2"><span class="lab"><font color="Green" title="E45">A24</font></span></a></span>;<br/><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><a NAME="E1:42_1_2_1_2_1"/><span class="lab"><font color="Green" title="E48">A27</font></span>: 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_1_2_1_2"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="ref" href="xxreal_0.html#D10" title="XXREAL_0:def.10">XXREAL_0:def 10</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_1_2_1_2_1_1"/>
(  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> or  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T16" title="XXREAL_0:th.16">XXREAL_0:16</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_2_1_2_1_1_1"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_1_2_1_2_1_1_1"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E6:42_1_2_1_2"><span class="lab"><font color="Green" title="E46">A25</font></span></a>, <a class="txt" href="#E1:42_1_2_1_2_1"><span class="lab"><font color="Green" title="E48">A27</font></span></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_2_1_2_1_1_2"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_1_2_1_2_1_1_2"/>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:42_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E1:42_1_2_1_2"><span class="lab"><font color="Green" title="E45">A24</font></span></a>, <a class="txt" href="#E7:42_1_2_1_2"><span class="lab"><font color="Green" title="E47">A26</font></span></a>, <a class="txt" href="#E1:42_1_2_1_2_1"><span class="lab"><font color="Green" title="E48">A27</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E16:42_1"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">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>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span>]</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:42_1_3"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:42_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>;<br/>
<a NAME="E2:42_1_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E45">A28</font></span>: <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>) = 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E26">A2</font></span></a></span>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:42_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/>
<a NAME="E4:42_1_3"/><span class="kw">then </span>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>) <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E5:42_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A29</font></span>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:42_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_3_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:42_1_3_2"/>
( <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> or <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_3_2_1"><span class="kw">suppose </span></a><a NAME="E1:42_1_3_2_1"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">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>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>)</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:42_1_3_2_1"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">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>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E5:42_1_3"><span class="lab"><font color="Green" title="E46">A29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_3_2_2"><span class="kw">suppose </span></a><a NAME="E1:42_1_3_2_2"/>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">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>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>)</span><br/><div class="add"><span class="kw">then </span><font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span>) = 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E26">A2</font></span></a></span>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p4">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42_1"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42_1"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:42_1_3_2_2"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">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>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42_1"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E3:42_1_3"><span class="lab"><font color="Green" title="E45">A28</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E17:42_1"/><span class="lab"><font color="Green" title="E45">A30</font></span>: 
<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E8:42"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E2:42_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E9:42_1"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="txt" href="#E10:42_1"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E11:42_1"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E12:42_1"><span class="lab"><font color="Green" title="E43">A19</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E18:42_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c33">v</font> =  <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c33">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> &amp; <font color="Maroon" title="c33">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c33">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c33">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E1:42_1_4"/>
<font color="Maroon" title="c33">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c33">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c33">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c33">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E2:42_1_4"/>
( <font color="Maroon" title="c33">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c33">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:42_1"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c33">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c33">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:42_1"><span class="lab"><font color="Green" title="E45">A30</font></span></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c33">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c33">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c33">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E10:42"/><span class="lab"><font color="Green" title="E33">A31</font></span>: 
for <font color="Olive" title="b1">t</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a>   st <font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Olive" title="b1">t</font>] &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Olive" title="b1">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/><font color="Olive" title="b1">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font></span>)</span>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c26">t</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>] &amp; <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font></span>)</span>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:42_2"/><span class="lab"><font color="Green" title="E34">A32</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>]
 <span class="kw">and </span><br/><a NAME="E2:42_2"/><span class="lab"><font color="Green" title="E35">A33</font></span>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font></span>)</span>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<span class="kw">consider </span><font color="Maroon" title="c27">v</font> being   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmpds_2.html#K1" title="SCMPDS_2:func.1">SCMPDS</a> <span class="kw"> such that </span><br/><a NAME="E4:42_2"/><span class="lab"><font color="Green" title="E36">A34</font></span>: 
<font color="Maroon" title="c27">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>
 <span class="kw">and </span><br/><a NAME="E5:42_2"/><span class="lab"><font color="Green" title="E37">A35</font></span>: 
<font color="Maroon" title="c27">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E6:42_2"/>
<font color="Maroon" title="c27">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E7:42_2"/><span class="lab"><font color="Green" title="E38">A36</font></span>: 
<font color="Maroon" title="c27">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42_2"><span class="lab"><font color="Green" title="E34">A32</font></span></a></span>;<br/>
<div><span class="lab"><font color="Green" title="E39">A37</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:42_2_1"/>
<font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:42_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>;<br/><a NAME="E3:42_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A38</font></span>: <font color="Maroon">H<sub>1</sub></font>( <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font>) = 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:42"><span class="lab"><font color="Green" title="E26">A2</font></span></a></span>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c26">t</font></span>)</span> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>
<br/>.= 
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>
;<br/><a NAME="E5:42_2_1"/>
<font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_2"><span class="lab"><font color="Green" title="E36">A34</font></span></a>, <a class="txt" href="#E5:42_2"><span class="lab"><font color="Green" title="E37">A35</font></span></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>;<br/><a NAME="E6:42_2_1"/><span class="kw">then </span>
 <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T133" title="COMPLEX1:th.133">COMPLEX1:133</a></span>;<br/><span class="kw">hence </span><a NAME="E7:42_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:42_2"><span class="lab"><font color="Green" title="E35">A33</font></span></a>, <a class="txt" href="#E4:42_2_1"><span class="lab"><font color="Green" title="E40">A38</font></span></a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">thus </span><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font></span>)</span>,<font color="Maroon" title="c22">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E4:42_2"><span class="lab"><font color="Green" title="E36">A34</font></span></a>, <a class="txt" href="#E7:42_2"><span class="lab"><font color="Green" title="E38">A36</font></span></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">v</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_2"><span class="lab"><font color="Green" title="E36">A34</font></span></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:42_2"><span class="lab"><font color="Green" title="E36">A34</font></span></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:42_2"><span class="lab"><font color="Green" title="E39">A37</font></span></a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:42"/><span class="lab"><font color="Green" title="E34">A39</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c17">s</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span><font color="Maroon" title="c26">t</font> =  <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c17">s</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c17">s</font> &amp; <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E1:42_3"/>
<font color="Maroon" title="c26">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_8.html#K1" title="SCMPDS_8:func.1">Dstate</a> <font color="Maroon" title="c17">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E2:42_3"/>
<font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:42"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span> )</span><br/>

<span class="kw">thus </span><a NAME="E3:42_3"/>
<font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <font color="Maroon" title="c23">d</font>,<font color="Maroon" title="c22">i</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:42"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c20">b</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">t</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c21">c</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scmpds_8.html#T4" title="SCMPDS_8:th.4">SCMPDS_8:4</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:42"/>
(  <a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c17">s</font> &amp;  <a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c17">s</font> )
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="scpinvar.html#S3" target="_self" title="SCPINVAR:sch.3">SCPINVAR:sch 3</a>(<a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E33">A31</font></span></a>, <a class="txt" href="#E11:42"><span class="lab"><font color="Green" title="E34">A39</font></span></a>, <a class="txt" href="#E9:42"><span class="lab"><font color="Green" title="E32">A8</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E13:42"/>
(  <a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R1" title="SCMPDS_6:pred.1">is_closed_on</a> <font color="Maroon" title="c17">s</font> &amp;  <a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font> <a href="scmpds_6.html#R2" title="SCMPDS_6:pred.2">is_halting_on</a> <font color="Maroon" title="c17">s</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font></span>)</span>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c17">s</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E14:42"/><span class="lab"><font color="Green" title="E35">A40</font></span>: 
<font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <span class="p1">(<span class="default"><a href="scmpds_2.html#K3" title="SCMPDS_2:func.3">DataLoc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">s</font> <a href="scmpds_2.html#K2" title="SCMPDS_2:func.2">.</a> <font color="Maroon" title="c19">a</font></span>)</span>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c17">s</font></span>)</span></span><br/>

<a NAME="E15:42"/>
 <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c17">s</font></span>)</span>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="scpinvar.html#S4" target="_self" title="SCPINVAR:sch.4">SCPINVAR:sch 4</a>(<a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E14:42"><span class="lab"><font color="Green" title="E35">A40</font></span></a>, <a class="txt" href="#E10:42"><span class="lab"><font color="Green" title="E33">A31</font></span></a>, <a class="txt" href="#E11:42"><span class="lab"><font color="Green" title="E34">A39</font></span></a>, <a class="txt" href="#E9:42"><span class="lab"><font color="Green" title="E32">A8</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E16:42"/>
 <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <span class="p1">(<span class="default"><a href="scpinvar.html#K4" title="SCPINVAR:func.4">while&lt;&gt;0</a> <font color="Maroon" title="c19">a</font>,<font color="Maroon" title="c22">i</font>,<font color="Maroon" title="c18">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="scmpds_4.html#K9" title="SCMPDS_4:func.9">IExec</a> <font color="Maroon" title="c18">I</font>,<font color="Maroon" title="c17">s</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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