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<b>let </b><font color="Maroon">IIG</font> be  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V6">finite</a> non <a href="msualg_1.html#V2">void</a> <a href="msafree2.html#V2">Circuit-like</a> <a href="msafree2.html#V5">monotonic</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ;<br/><b>let </b><font color="Maroon">SCS</font> be  <a href="msualg_1.html#V5">non-empty</a> <a href="circuit1.html#NM1">Circuit</a> of <font color="Maroon">IIG</font>;<br/><b>let </b><font color="Maroon">InpFs</font> be   <a href="circuit1.html#NM2">InputFuncs</a> of <font color="Maroon">SCS</font>;<br/>





<b>assume </b><a NAME="E1:26"/><i><font color="Green">E42</font></i>: 
(  <a href="funct_6.html#K10">commute</a> <font color="Maroon">InpFs</font> is <a href="funct_1.html#V3">constant</a> &amp; not  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/>

<a NAME="E2:26"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">IIG</font> is <a href="msafree2.html#V1">with_input_V</a>
 
<b>by </b><i><a class="ref" href="msafree2.html#D4">MSAFREE2:def 4</a></i>;<br/>
<b>let </b><font color="Maroon">s</font> be   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">SCS</font>;<br/><b>let </b><font color="Maroon">iv</font> be    <a href="msafree2.html#M1">InputValues</a> of <font color="Maroon">SCS</font>;<br/>



<b>assume </b><a NAME="E3:26"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">iv</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">InpFs</font></span>)</span> <a href="funct_1.html#K1">.</a> 0
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">v</font> being  <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font>  st  <a href="circuit1.html#K6">depth</a> <font color="Olive">v</font>,<font color="Maroon">SCS</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K9" target="_self">Computation</a> <font color="Maroon">s</font>,<font color="Maroon">InpFs</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a>  <a href="circuit2.html#K5" target="_self">IGValue</a> <font color="Olive">v</font>,<font color="Maroon">iv</font>;<br/>
<a NAME="E4:26"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">v</font> be   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font>;<br/>

<b>assume </b><a NAME="E1:26_1"/>
 <a href="circuit1.html#K6">depth</a> <font color="Maroon">v</font>,<font color="Maroon">SCS</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:26_1"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="circuit1.html#K6">depth</a> <font color="Maroon">v</font>,<font color="Maroon">SCS</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a></i>;<br/>
<i><font color="Green">E48</font></i>: <span class="p1">(<span class="default"><a href="circuit2.html#K9" target="_self">Computation</a> <font color="Maroon">s</font>,<font color="Maroon">InpFs</font></span>)</span> <a href="funct_2.html#K3">.</a> 0 = 
 <a href="circuit2.html#K8" target="_self">InitialComp</a> <font color="Maroon">s</font>,<font color="Maroon">InpFs</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="circuit2.html#K1" target="_self">+*</a> <span class="p2">(<span class="default">0 <a href="circuit1.html#K2">-th_InputValues</a> <font color="Maroon">InpFs</font></span>)</span></span>)</span> <a href="funct_4.html#K1">+*</a> <span class="p1">(<span class="default"><a href="circuit1.html#K1">Set-Constants</a> <font color="Maroon">SCS</font></span>)</span>

;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_2"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font> or <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font> )
 </span><b>by </b><i>, <a class="ref" href="circuit1.html#T19">CIRCUIT1:19</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2_1"><b>suppose </b></a><a NAME="E1:26_1_2_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font>
 ;<br/><div class="add"><a NAME="E2:26_1_2_1"/>
 <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font> <a href="xboole_0.html#R1">misses</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font>
 <b>by </b><i><a class="ref" href="msafree2.html#T6">MSAFREE2:6</a></i>;<br/><a NAME="E3:26_1_2_1"/><b>then </b>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><a NAME="E4:26_1_2_1"/><b>then </b><i><font color="Green">E51</font></i>: 
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="circuit1.html#K1">Set-Constants</a> <font color="Maroon">SCS</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><a NAME="E5:26_1_2_1"/><i><font color="Green">E52</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">0 <a href="circuit1.html#K2">-th_InputValues</a> <font color="Maroon">InpFs</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K9" target="_self">Computation</a> <font color="Maroon">s</font>,<font color="Maroon">InpFs</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon">s</font> <a href="circuit2.html#K1" target="_self">+*</a> <span class="p2">(<span class="default">0 <a href="circuit1.html#K2">-th_InputValues</a> <font color="Maroon">InpFs</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default">0 <a href="circuit1.html#K2">-th_InputValues</a> <font color="Maroon">InpFs</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i>, , <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon">iv</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i>, , <a class="ref" href="circuit1.html#D2">CIRCUIT1:def 2</a></i>
<br/>.= 
 <a href="circuit2.html#K5" target="_self">IGValue</a> <font color="Maroon">v</font>,<font color="Maroon">iv</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_2_2"><b>suppose </b></a><a NAME="E1:26_1_2_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K1">SortsWithConstants</a> <font color="Maroon">IIG</font>
 ;<br/><div class="add"><a NAME="E2:26_1_2_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="circuit1.html#K1">Set-Constants</a> <font color="Maroon">SCS</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K9" target="_self">Computation</a> <font color="Maroon">s</font>,<font color="Maroon">InpFs</font></span>)</span> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><a href="circuit1.html#K1">Set-Constants</a> <font color="Maroon">SCS</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i>, <a class="ref" href="funct_4.html#T14">FUNCT_4:14</a></i>
<br/>.= 
 <a href="circuit2.html#K5" target="_self">IGValue</a> <font color="Maroon">v</font>,<font color="Maroon">iv</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:26"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>reconsider </b><font color="Maroon">Ck</font> = <span class="p1">(<span class="default"><a href="circuit2.html#K9" target="_self">Computation</a> <font color="Maroon">s</font>,<font color="Maroon">InpFs</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> as   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">SCS</font> ;<br/>
<b>assume </b><a NAME="E2:26_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E3:26_2"/><i><font color="Green">E61</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">InpFs</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i>, <a class="ref" href="pre_circ.html#T8">PRE_CIRC:8</a></i>;<br/>
<i><font color="Green">E62</font></i>: <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="circuit1.html#K2">-th_InputValues</a> <font color="Maroon">InpFs</font> = 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">InpFs</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="circuit1.html#D2">CIRCUIT1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_6.html#K10">commute</a> <font color="Maroon">InpFs</font></span>)</span> <a href="funct_1.html#K1">.</a> 0
<b>by </b><i>, , <a class="ref" href="funct_1.html#D16">FUNCT_1:def 16</a></i>
;<br/>
<a NAME="E5:26_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">iv</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Ck</font>
 
<b>by </b><i>, , </i>;<br/>
<b>let </b><font color="Maroon">v</font> be   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font>;<br/>

<b>assume </b><a NAME="E6:26_2"/><i><font color="Green">E65</font></i>: 
 <a href="circuit1.html#K6">depth</a> <font color="Maroon">v</font>,<font color="Maroon">SCS</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K9" target="_self">Computation</a> <font color="Maroon">s</font>,<font color="Maroon">InpFs</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> = 
<span class="p1">(<span class="default"><a href="circuit2.html#K7" target="_self">Following</a> <font color="Maroon">Ck</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="circuit1.html#K2">-th_InputValues</a> <font color="Maroon">InpFs</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="circuit2.html#K6" target="_self">Following</a> <font color="Maroon">Ck</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
<b>by </b><i>, , , <a class="ref" href="funct_4.html#T79">FUNCT_4:79</a></i>
<br/>.= 
 <a href="circuit2.html#K5" target="_self">IGValue</a> <font color="Maroon">v</font>,<font color="Maroon">iv</font>
<b>by </b><i>, <a class="txt" href="circuit2.html#E1"><i><font color="Green">Lemma32</font></i></a>, </i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E6:26"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


</div>
