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<span class="kw">let </span><font color="Maroon" title="c1">ap</font>, <font color="Maroon" title="c2">bp</font>, <font color="Maroon" title="c3">cp</font>, <font color="Maroon" title="c4">dp</font> be  non  <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">cin</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">cin</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">dp</font>,<span class="p3">(<span class="default"><a href="gfacirc1.html#K12" title="GFACIRC1:func.12">GFA0AdderOutput</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; not <font color="Olive" title="b1">cin</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K13" title="GFACIRC1:func.13">BitGFA0Str</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span> holds <br/>for <font color="Olive" title="b2">s</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><a href="ftacell1.html#K2" title="FTACELL1:func.2">BitFTA0Circ</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font>,<font color="Maroon" title="c4">dp</font>,<font color="Olive" title="b1">cin</font></span>)</span><br/> for <font color="Olive" title="b3">a1</font>, <font color="Olive" title="b4">a2</font>, <font color="Olive" title="b5">a3</font>, <font color="Olive" title="b6">a4</font>, <font color="Olive" title="b7">a5</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="circcomb.html#K8" title="CIRCCOMB:func.8">BOOLEAN</a>   st <font color="Olive" title="b3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ap</font> &amp; <font color="Olive" title="b4">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">bp</font> &amp; <font color="Olive" title="b5">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">cp</font> &amp; <font color="Olive" title="b6">a4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">dp</font> &amp; <font color="Olive" title="b7">a5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">cin</font> holds <br/>( <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Olive" title="b2">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K12" title="GFACIRC1:func.12">GFA0AdderOutput</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">a1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Olive" title="b4">a2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Olive" title="b5">a3</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Olive" title="b2">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ap</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a1</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Olive" title="b2">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">bp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">a2</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Olive" title="b2">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">cp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">a3</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Olive" title="b2">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">dp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">a4</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Olive" title="b2">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">cin</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b7">a5</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">cin</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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="c5">cin</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">dp</font>,<span class="p3">(<span class="default"><a href="gfacirc1.html#K12" title="GFACIRC1:func.12">GFA0AdderOutput</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; not <font color="Maroon" title="c5">cin</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K13" title="GFACIRC1:func.13">BitGFA0Str</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span> implies for <font color="Olive" title="b1">s</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><a href="ftacell1.html#K2" title="FTACELL1:func.2">BitFTA0Circ</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font>,<font color="Maroon" title="c4">dp</font>,<font color="Maroon" title="c5">cin</font></span>)</span><br/> for <font color="Olive" title="b2">a1</font>, <font color="Olive" title="b3">a2</font>, <font color="Olive" title="b4">a3</font>, <font color="Olive" title="b5">a4</font>, <font color="Olive" title="b6">a5</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="circcomb.html#K8" title="CIRCCOMB:func.8">BOOLEAN</a>   st <font color="Olive" title="b2">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ap</font> &amp; <font color="Olive" title="b3">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">bp</font> &amp; <font color="Olive" title="b4">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">cp</font> &amp; <font color="Olive" title="b5">a4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">dp</font> &amp; <font color="Olive" title="b6">a5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">cin</font> holds <br/>( <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Olive" title="b1">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K12" title="GFACIRC1:func.12">GFA0AdderOutput</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Olive" title="b3">a2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Olive" title="b4">a3</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Olive" title="b1">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ap</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a1</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Olive" title="b1">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">bp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a2</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Olive" title="b1">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">cp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">a3</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Olive" title="b1">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">dp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">a4</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Olive" title="b1">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">cin</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">a5</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:13"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
( <font color="Maroon" title="c5">cin</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">dp</font>,<span class="p3">(<span class="default"><a href="gfacirc1.html#K12" title="GFACIRC1:func.12">GFA0AdderOutput</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>,<a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; not <font color="Maroon" title="c5">cin</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K13" title="GFACIRC1:func.13">BitGFA0Str</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span> )
 ; <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">s</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><a href="ftacell1.html#K2" title="FTACELL1:func.2">BitFTA0Circ</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font>,<font color="Maroon" title="c4">dp</font>,<font color="Maroon" title="c5">cin</font></span>)</span><br/> for <font color="Olive" title="b2">a1</font>, <font color="Olive" title="b3">a2</font>, <font color="Olive" title="b4">a3</font>, <font color="Olive" title="b5">a4</font>, <font color="Olive" title="b6">a5</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="circcomb.html#K8" title="CIRCCOMB:func.8">BOOLEAN</a>   st <font color="Olive" title="b2">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ap</font> &amp; <font color="Olive" title="b3">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">bp</font> &amp; <font color="Olive" title="b4">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">cp</font> &amp; <font color="Olive" title="b5">a4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">dp</font> &amp; <font color="Olive" title="b6">a5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">cin</font> holds <br/>( <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Olive" title="b1">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K12" title="GFACIRC1:func.12">GFA0AdderOutput</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Olive" title="b3">a2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Olive" title="b4">a3</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Olive" title="b1">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ap</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a1</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Olive" title="b1">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">bp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a2</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Olive" title="b1">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">cp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">a3</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Olive" title="b1">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">dp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">a4</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Olive" title="b1">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">cin</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">a5</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">A1</font> =  <a href="gfacirc1.html#K12" title="GFACIRC1:func.12">GFA0AdderOutput</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">C1</font> =  <a href="gfacirc1.html#K14" title="GFACIRC1:func.14">BitGFA0Circ</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">S1</font> =  <a href="gfacirc1.html#K13" title="GFACIRC1:func.13">BitGFA0Str</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">S2</font> =  <a href="gfacirc1.html#K13" title="GFACIRC1:func.13">BitGFA0Str</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K12" title="GFACIRC1:func.12">GFA0AdderOutput</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span>,<font color="Maroon" title="c5">cin</font>,<font color="Maroon" title="c4">dp</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">C2</font> =  <a href="gfacirc1.html#K14" title="GFACIRC1:func.14">BitGFA0Circ</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K12" title="GFACIRC1:func.12">GFA0AdderOutput</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span>,<font color="Maroon" title="c5">cin</font>,<font color="Maroon" title="c4">dp</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">S</font> =  <a href="ftacell1.html#K1" title="FTACELL1:func.1">BitFTA0Str</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font>,<font color="Maroon" title="c4">dp</font>,<font color="Maroon" title="c5">cin</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c12">s</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><a href="ftacell1.html#K2" title="FTACELL1:func.2">BitFTA0Circ</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font>,<font color="Maroon" title="c4">dp</font>,<font color="Maroon" title="c5">cin</font></span>)</span>; <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">a1</font>, <font color="Olive" title="b2">a2</font>, <font color="Olive" title="b3">a3</font>, <font color="Olive" title="b4">a4</font>, <font color="Olive" title="b5">a5</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="circcomb.html#K8" title="CIRCCOMB:func.8">BOOLEAN</a>   st <font color="Olive" title="b1">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ap</font> &amp; <font color="Olive" title="b2">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">bp</font> &amp; <font color="Olive" title="b3">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">cp</font> &amp; <font color="Olive" title="b4">a4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">dp</font> &amp; <font color="Olive" title="b5">a5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">cin</font> holds <br/>( <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K12" title="GFACIRC1:func.12">GFA0AdderOutput</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Olive" title="b2">a2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Olive" title="b3">a3</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ap</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a1</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">bp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a2</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">cp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">a3</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">dp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">a4</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">cin</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">a5</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c13">a1</font>, <font color="Maroon" title="c14">a2</font>, <font color="Maroon" title="c15">a3</font>, <font color="Maroon" title="c16">a4</font>, <font color="Maroon" title="c17">a5</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="circcomb.html#K8" title="CIRCCOMB:func.8">BOOLEAN</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="c13">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ap</font> &amp; <font color="Maroon" title="c14">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">bp</font> &amp; <font color="Maroon" title="c15">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">cp</font> &amp; <font color="Maroon" title="c16">a4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">dp</font> &amp; <font color="Maroon" title="c17">a5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">cin</font> implies ( <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K12" title="GFACIRC1:func.12">GFA0AdderOutput</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c14">a2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c15">a3</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ap</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a1</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">bp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a2</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">cp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a3</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">dp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a4</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">cin</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a5</font> ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:13"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
<font color="Maroon" title="c13">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ap</font>
 <span class="kw">and </span><br/><a NAME="E3:13"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c14">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">bp</font>
 <span class="kw">and </span><br/><a NAME="E4:13"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c15">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">cp</font>
 <span class="kw">and </span><br/><a NAME="E5:13"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
<font color="Maroon" title="c16">a4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">dp</font>
 <span class="kw">and </span><br/><a NAME="E6:13"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<font color="Maroon" title="c17">a5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">cin</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="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K12" title="GFACIRC1:func.12">GFA0AdderOutput</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c14">a2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c15">a3</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ap</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a1</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">bp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a2</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">cp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a3</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">dp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a4</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">cin</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a5</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c18">s1</font> = <font color="Maroon" title="c12">s</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K13" title="GFACIRC1:func.13">BitGFA0Str</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span> as   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K14" title="GFACIRC1:func.14">BitGFA0Circ</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T26" title="FACIRC_1:th.26">FACIRC_1:26</a></span>;<br/>
<a NAME="E8:13"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c18">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K13" title="GFACIRC1:func.13">BitGFA0Str</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="circuit1.html#T4" title="CIRCUIT1:th.4">CIRCUIT1:4</a></span>;<br/>
<a NAME="E9:13"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c4">dp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="ftacell1.html#K1" title="FTACELL1:func.1">BitFTA0Str</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font>,<font color="Maroon" title="c4">dp</font>,<font color="Maroon" title="c5">cin</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="ftacell1.html#T6" target="_self" title="FTACELL1:th.6">Th6</a></span>;<br/>
<a NAME="E10:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">dp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:13"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="circuit2.html#D5" title="CIRCUIT2:def.5">CIRCUIT2:def 5</a></span>;<br/>
<a NAME="E11:13"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c3">cp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="ftacell1.html#K1" title="FTACELL1:func.1">BitFTA0Str</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font>,<font color="Maroon" title="c4">dp</font>,<font color="Maroon" title="c5">cin</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="ftacell1.html#T6" target="_self" title="FTACELL1:th.6">Th6</a></span>;<br/>
<a NAME="E12:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">cp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="circuit2.html#D5" title="CIRCUIT2:def.5">CIRCUIT2:def 5</a></span>;<br/>
<a NAME="E13:13"/>
<font color="Maroon" title="c2">bp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K13" title="GFACIRC1:func.13">BitGFA0Str</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gfacirc1.html#T44" title="GFACIRC1:th.44">GFACIRC1:44</a></span>;<br/>
<a NAME="E14:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A12</font></span>: 
<font color="Maroon" title="c14">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">bp</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E8:13"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">t</font> = <font color="Maroon" title="c12">s</font> as   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gfacirc1.html#K14" title="GFACIRC1:func.14">BitGFA0Circ</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <span class="p2">(<span class="default"><a href="gfacirc1.html#K14" title="GFACIRC1:func.14">BitGFA0Circ</a> <span class="p3">(<span class="default"><a href="gfacirc1.html#K12" title="GFACIRC1:func.12">GFA0AdderOutput</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span>,<font color="Maroon" title="c5">cin</font>,<font color="Maroon" title="c4">dp</font></span>)</span></span>)</span> ;<br/>
<a NAME="E16:13"/>
(  <a href="gfacirc1.html#K12" title="GFACIRC1:func.12">GFA0AdderOutput</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K13" title="GFACIRC1:func.13">BitGFA0Str</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span> &amp;  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K13" title="GFACIRC1:func.13">BitGFA0Str</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K13" title="GFACIRC1:func.13">BitGFA0Str</a> <span class="p2">(<span class="default"><a href="gfacirc1.html#K12" title="GFACIRC1:func.12">GFA0AdderOutput</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span>,<font color="Maroon" title="c5">cin</font>,<font color="Maroon" title="c4">dp</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="ftacell1.html#E6"><span class="lab"><font color="Green" title="E3">Lm2</font></span></a>, <a class="ref" href="gfacirc1.html#T44" title="GFACIRC1:th.44">GFACIRC1:44</a></span>;<br/>
<a NAME="E17:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A13</font></span>: 
<span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c19">t</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K12" title="GFACIRC1:func.12">GFA0AdderOutput</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c18">s1</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K12" title="GFACIRC1:func.12">GFA0AdderOutput</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T32" title="FACIRC_1:th.32">FACIRC_1:32</a></span>;<br/>
<a NAME="E18:13"/>
<font color="Maroon" title="c3">cp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K13" title="GFACIRC1:func.13">BitGFA0Str</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gfacirc1.html#T44" title="GFACIRC1:th.44">GFACIRC1:44</a></span>;<br/>
<a NAME="E19:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A14</font></span>: 
<font color="Maroon" title="c15">a3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">cp</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:13"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E8:13"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/>
<a NAME="E20:13"/>
<font color="Maroon" title="c1">ap</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="gfacirc1.html#K13" title="GFACIRC1:func.13">BitGFA0Str</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gfacirc1.html#T44" title="GFACIRC1:th.44">GFACIRC1:44</a></span>;<br/>
<a NAME="E21:13"/><span class="kw">then </span>
<font color="Maroon" title="c13">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">s1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ap</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E8:13"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/>
<span class="kw">hence </span><a NAME="E22:13"/>
<span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="gfacirc1.html#K12" title="GFACIRC1:func.12">GFA0AdderOutput</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">a1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c14">a2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c15">a3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:13"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E19:13"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E17:13"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="ref" href="gfacirc1.html#T47" title="GFACIRC1:th.47">GFACIRC1:47</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="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ap</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a1</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">bp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a2</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">cp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a3</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">dp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a4</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">cin</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a5</font> )</span><br/>

<a NAME="E23:13"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
<font color="Maroon" title="c2">bp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="ftacell1.html#K1" title="FTACELL1:func.1">BitFTA0Str</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font>,<font color="Maroon" title="c4">dp</font>,<font color="Maroon" title="c5">cin</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="ftacell1.html#T6" target="_self" title="FTACELL1:th.6">Th6</a></span>;<br/>
<a NAME="E24:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A16</font></span>: 
<span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">bp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:13"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="circuit2.html#D5" title="CIRCUIT2:def.5">CIRCUIT2:def 5</a></span>;<br/>
<a NAME="E25:13"/><span class="lab"><font color="Green" title="E25">A17</font></span>: 
<font color="Maroon" title="c5">cin</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="ftacell1.html#K1" title="FTACELL1:func.1">BitFTA0Str</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font>,<font color="Maroon" title="c4">dp</font>,<font color="Maroon" title="c5">cin</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="ftacell1.html#T6" target="_self" title="FTACELL1:th.6">Th6</a></span>;<br/>
<a NAME="E26:13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A18</font></span>: 
<span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">cin</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a5</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:13"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="circuit2.html#D5" title="CIRCUIT2:def.5">CIRCUIT2:def 5</a></span>;<br/>
<a NAME="E27:13"/><span class="lab"><font color="Green" title="E27">A19</font></span>: 
<font color="Maroon" title="c1">ap</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="ftacell1.html#K1" title="FTACELL1:func.1">BitFTA0Str</a> <font color="Maroon" title="c1">ap</font>,<font color="Maroon" title="c2">bp</font>,<font color="Maroon" title="c3">cp</font>,<font color="Maroon" title="c4">dp</font>,<font color="Maroon" title="c5">cin</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:13"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="ref" href="ftacell1.html#T6" target="_self" title="FTACELL1:th.6">Th6</a></span>;<br/>
<a NAME="E28:13"/><span class="kw">then </span>
(  <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c12">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <font color="Maroon" title="c12">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ap</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:13"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="circuit2.html#D5" title="CIRCUIT2:def.5">CIRCUIT2:def 5</a>, <a class="ref" href="facirc_1.html#T15" title="FACIRC_1:th.15">FACIRC_1:15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E29:13"/>
( <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">ap</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">a1</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">bp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a2</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">cp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">a3</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">dp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">a4</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">cin</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a5</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:13"><span class="lab"><font color="Green" title="E27">A19</font></span></a>, <a class="txt" href="#E23:13"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E11:13"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E9:13"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E25:13"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="txt" href="#E24:13"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="txt" href="#E12:13"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="txt" href="#E10:13"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E26:13"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="ref" href="circuit2.html#D5" title="CIRCUIT2:def.5">CIRCUIT2:def 5</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/>


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