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<b>let </b><font color="Maroon" title="c1">IIG</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V8" title="STRUCT_0:attr.8">finite</a> non <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a> <a href="msafree2.html#V2" title="MSAFREE2:attr.2">Circuit-like</a> <a href="msafree2.html#V5" title="MSAFREE2:attr.5">monotonic</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">SCS</font> being <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">IIG</font><br/> for <font color="Olive" title="b2">InpFs</font> being  <a href="circuit1.html#NM2" title="CIRCUIT1:NM.2">InputFuncs</a> of <font color="Olive" title="b1">SCS</font><br/> for <font color="Olive" title="b3">iv</font> being   <a href="msafree2.html#M1" title="MSAFREE2:mode.1">InputValues</a> of <font color="Olive" title="b1">SCS</font>  st  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Olive" title="b2">InpFs</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; not  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">IIG</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Olive" title="b3">iv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Olive" title="b2">InpFs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>for <font color="Olive" title="b4">s</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Olive" title="b1">SCS</font><br/> for <font color="Olive" title="b5">v</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font><br/> for <font color="Olive" title="b6">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b6">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit1.html#K7" title="CIRCUIT1:func.7">depth</a> <font color="Olive" title="b1">SCS</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Olive" title="b4">s</font>,<font color="Olive" title="b2">InpFs</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b6">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Olive" title="b5">v</font>,<font color="Olive" title="b3">iv</font></span><br/><b>let </b><font color="Maroon" title="c2">SCS</font> be  <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a> <a href="circuit1.html#NM1" title="CIRCUIT1:NM.1">Circuit</a> of <font color="Maroon" title="c1">IIG</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">InpFs</font> being  <a href="circuit1.html#NM2" title="CIRCUIT1:NM.2">InputFuncs</a> of <font color="Maroon" title="c2">SCS</font><br/> for <font color="Olive" title="b2">iv</font> being   <a href="msafree2.html#M1" title="MSAFREE2:mode.1">InputValues</a> of <font color="Maroon" title="c2">SCS</font>  st  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Olive" title="b1">InpFs</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; not  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">IIG</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Olive" title="b2">iv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Olive" title="b1">InpFs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>for <font color="Olive" title="b3">s</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c2">SCS</font><br/> for <font color="Olive" title="b4">v</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font><br/> for <font color="Olive" title="b5">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit1.html#K7" title="CIRCUIT1:func.7">depth</a> <font color="Maroon" title="c2">SCS</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Olive" title="b3">s</font>,<font color="Olive" title="b1">InpFs</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b5">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Olive" title="b4">v</font>,<font color="Olive" title="b2">iv</font></span><br/><b>let </b><font color="Maroon" title="c3">InpFs</font> be   <a href="circuit1.html#NM2" title="CIRCUIT1:NM.2">InputFuncs</a> of <font color="Maroon" title="c2">SCS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">iv</font> being   <a href="msafree2.html#M1" title="MSAFREE2:mode.1">InputValues</a> of <font color="Maroon" title="c2">SCS</font>  st  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c3">InpFs</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; not  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">IIG</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Olive" title="b1">iv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>for <font color="Olive" title="b2">s</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c2">SCS</font><br/> for <font color="Olive" title="b3">v</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font><br/> for <font color="Olive" title="b4">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit1.html#K7" title="CIRCUIT1:func.7">depth</a> <font color="Maroon" title="c2">SCS</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Olive" title="b2">s</font>,<font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b4">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Olive" title="b3">v</font>,<font color="Olive" title="b1">iv</font></span><br/><b>let </b><font color="Maroon" title="c4">iv</font> be    <a href="msafree2.html#M1" title="MSAFREE2:mode.1">InputValues</a> of <font color="Maroon" title="c2">SCS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c3">InpFs</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; not  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">IIG</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c4">iv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b1">s</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c2">SCS</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font><br/> for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit1.html#K7" title="CIRCUIT1:func.7">depth</a> <font color="Maroon" title="c2">SCS</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Olive" title="b2">v</font>,<font color="Maroon" title="c4">iv</font> )</span><br/>







<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E24">A1</font></i>: 
(  <a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c3">InpFs</font> is <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a> &amp; not  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <font color="Maroon" title="c1">IIG</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c4">iv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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 <font color="Maroon" title="c2">SCS</font><br/> for <font color="Olive" title="b2">v</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font><br/> for <font color="Olive" title="b3">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit1.html#K7" title="CIRCUIT1:func.7">depth</a> <font color="Maroon" title="c2">SCS</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Olive" title="b2">v</font>,<font color="Maroon" title="c4">iv</font></span><br/>

<b>let </b><font color="Maroon" title="c5">s</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <font color="Maroon" title="c2">SCS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">v</font> being  <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font><br/> for <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit1.html#K7" title="CIRCUIT1:func.7">depth</a> <font color="Maroon" title="c2">SCS</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Olive" title="b1">v</font>,<font color="Maroon" title="c4">iv</font></span><br/><b>let </b><font color="Maroon" title="c6">v</font> be   <a href="msafree2.html#NM1" title="MSAFREE2:NM.1">Vertex</a> of <font color="Maroon" title="c1">IIG</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit1.html#K7" title="CIRCUIT1:func.7">depth</a> <font color="Maroon" title="c2">SCS</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c4">iv</font></span><br/><b>let </b><font color="Maroon" title="c7">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit1.html#K7" title="CIRCUIT1:func.7">depth</a> <font color="Maroon" title="c2">SCS</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c4">iv</font> )</span><br/>





<b>assume </b><a NAME="E2:27"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit1.html#K7" title="CIRCUIT1:func.7">depth</a> <font color="Maroon" title="c2">SCS</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c4">iv</font></span><br/>

<a NAME="E3:27"/>
 <a href="circuit1.html#K6" title="CIRCUIT1:func.6">depth</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c2">SCS</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="circuit1.html#K7" title="CIRCUIT1:func.7">depth</a> <font color="Maroon" title="c2">SCS</font>
 
<b>by </b><i><a class="ref" href="circuit1.html#T18" title="CIRCUIT1:th.18">CIRCUIT1:18</a></i>;<br/>
<b>hence </b><a NAME="E4:27"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="circuit2.html#K9" title="CIRCUIT2:func.9">Computation</a> <font color="Maroon" title="c5">s</font>,<font color="Maroon" title="c3">InpFs</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K5" title="CIRCUIT2:func.5">IGValue</a> <font color="Maroon" title="c6">v</font>,<font color="Maroon" title="c4">iv</font>
 <b>by </b><i><a class="txt" href="#E1:27"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="#E2:27"><i><font color="Green" title="E25">A2</font></i></a>, <a class="ref" href="circuit2.html#T16" target="_self" title="CIRCUIT2:th.16">Th16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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