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<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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">c</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K8" title="CIRCCOMB:func.8">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> holds <br/>( 1 <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> <a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K8" title="CIRCCOMB:func.8">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span></span>)</span> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K19" title="FACIRC_1:func.19">BitAdderWithOverflowStr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<font color="Olive" title="b1">c</font></span>)</span> &amp; 1 <a href="facirc_2.html#K5" title="FACIRC_2:func.5">-BitAdderCirc</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> <a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K8" title="CIRCCOMB:func.8">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span></span>)</span> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K20" title="FACIRC_1:func.20">BitAdderWithOverflowCirc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<font color="Olive" title="b1">c</font></span>)</span> &amp; 1 <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="facirc_1.html#K17" title="FACIRC_1:func.17">MajorityOutput</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<font color="Olive" title="b1">c</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">c</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="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K8" title="CIRCCOMB:func.8">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> implies ( 1 <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> <a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K8" title="CIRCCOMB:func.8">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span></span>)</span> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K19" title="FACIRC_1:func.19">BitAdderWithOverflowStr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c3">c</font></span>)</span> &amp; 1 <a href="facirc_2.html#K5" title="FACIRC_2:func.5">-BitAdderCirc</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> <a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K8" title="CIRCCOMB:func.8">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span></span>)</span> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K20" title="FACIRC_1:func.20">BitAdderWithOverflowCirc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c3">c</font></span>)</span> &amp; 1 <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="facirc_1.html#K17" title="FACIRC_1:func.17">MajorityOutput</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c3">c</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:19"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K8" title="CIRCCOMB:func.8">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> <a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K8" title="CIRCCOMB:func.8">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span></span>)</span> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K19" title="FACIRC_1:func.19">BitAdderWithOverflowStr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c3">c</font></span>)</span> &amp; 1 <a href="facirc_2.html#K5" title="FACIRC_2:func.5">-BitAdderCirc</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> <a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K8" title="CIRCCOMB:func.8">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span></span>)</span> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K20" title="FACIRC_1:func.20">BitAdderWithOverflowCirc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c3">c</font></span>)</span> &amp; 1 <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="facirc_1.html#K17" title="FACIRC_1:func.17">MajorityOutput</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c3">c</font> )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font>, <font color="Maroon" title="c6">h</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:19"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
1 <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">and </span><br/><a NAME="E4:19"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
1 <a href="facirc_2.html#K5" title="FACIRC_2:func.5">-BitAdderCirc</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">and </span><br/><a NAME="E5:19"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> <a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K8" title="CIRCCOMB:func.8">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:19"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> <a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K8" title="CIRCCOMB:func.8">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:19"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
<font color="Maroon" title="c6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 <span class="kw">and </span><br/><a NAME="E8:19"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><br/> for <font color="Olive" title="b2">S</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> <br/> for <font color="Olive" title="b3">A</font> being  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Olive" title="b2">S</font><br/> for <font color="Olive" title="b4">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> holds <br/>( <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">S</font> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K19" title="FACIRC_1:func.19">BitAdderWithOverflowStr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Olive" title="b4">z</font></span>)</span> &amp; <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A</font> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K20" title="FACIRC_1:func.20">BitAdderWithOverflowCirc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Olive" title="b4">z</font></span>)</span> &amp; <font color="Maroon" title="c6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="facirc_1.html#K17" title="FACIRC_1:func.17">MajorityOutput</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Olive" title="b4">z</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="facirc_2.html#D4" target="_self" title="FACIRC_2:def.4">Def4</a></span>;<br/>
<a NAME="E9:19"/>
1 <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E6:19"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E7:19"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E8:19"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="facirc_2.html#T7" target="_self" title="FACIRC_2:th.7">Th7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:19"/>
( 1 <a href="facirc_2.html#K4" title="FACIRC_2:func.4">-BitAdderStr</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="circcomb.html#K5" title="CIRCCOMB:func.5">1GateCircStr</a> <a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K8" title="CIRCCOMB:func.8">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span></span>)</span> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K19" title="FACIRC_1:func.19">BitAdderWithOverflowStr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c3">c</font></span>)</span> &amp; 1 <a href="facirc_2.html#K5" title="FACIRC_2:func.5">-BitAdderCirc</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7" title="CIRCCOMB:func.7">1GateCircuit</a> <a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K8" title="CIRCCOMB:func.8">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span>)</span></span>)</span> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K20" title="FACIRC_1:func.20">BitAdderWithOverflowCirc</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c3">c</font></span>)</span> &amp; 1 <a href="facirc_2.html#K6" title="FACIRC_2:func.6">-BitMajorityOutput</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="facirc_1.html#K17" title="FACIRC_1:func.17">MajorityOutput</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">b</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>,<font color="Maroon" title="c3">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E6:19"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E7:19"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E8:19"><span class="lab"><font color="Green" title="E16">A7</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/>


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