<?xml version="1.0"?>
<div class="add">

<a NAME="E1:36"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
for <font color="Olive" title="b1">nt</font> being  <a href="dtconstr.html#NM3" title="DTCONSTR:NM.3">NonTerminal</a> of <a href="scm_comp.html#K1" title="SCM_COMP:func.1">SCM-AE</a> <br/> for <font color="Olive" title="b2">tl</font>, <font color="Olive" title="b3">tr</font> being  <a href="scm_comp.html#NM1" title="SCM_COMP:NM.1">bin-term</a>  st <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Olive" title="b2">tl</font></span>)</span>,<span class="p2">(<span class="default"><a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Olive" title="b3">tr</font></span>)</span></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">tl</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b3">tr</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Olive" title="b2">tl</font>,<font color="Olive" title="b3">tr</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">nt</font> be   <a href="dtconstr.html#NM3" title="DTCONSTR:NM.3">NonTerminal</a> of <a href="scm_comp.html#K1" title="SCM_COMP:func.1">SCM-AE</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">tl</font>, <font color="Olive" title="b2">tr</font> being  <a href="scm_comp.html#NM1" title="SCM_COMP:NM.1">bin-term</a>  st <font color="Maroon" title="c11">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Olive" title="b1">tl</font></span>)</span>,<span class="p2">(<span class="default"><a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Olive" title="b2">tr</font></span>)</span></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">tl</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">tr</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Olive" title="b1">tl</font>,<font color="Olive" title="b2">tr</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c12">tl</font>, <font color="Maroon" title="c13">tr</font> be   <a href="scm_comp.html#NM1" title="SCM_COMP:NM.1">bin-term</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="c11">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c12">tl</font></span>)</span>,<span class="p2">(<span class="default"><a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c13">tr</font></span>)</span></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">tl</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">tr</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font>] )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:36_1"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c11">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c12">tl</font></span>)</span>,<span class="p2">(<span class="default"><a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c13">tr</font></span>)</span></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E2:36_1"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">tl</font>]
 <span class="kw">and </span><br/><a NAME="E3:36_1"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c13">tr</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c14">aux</font>, <font color="Maroon" title="c15">n</font>, <font color="Maroon" title="c16">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</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">s</font> being   <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c15">n</font>,<font color="Maroon" title="c15">n</font>,<font color="Maroon" title="c16">k</font>, <a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font>, <a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Maroon" title="c14">aux</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="scm_comp.html#K21" title="SCM_COMP:func.21">max_Data-Loc_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  ex <font color="Olive" title="b3">u</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  st <br/>( <font color="Olive" title="b3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Olive" title="b1">s</font></span>)</span>,<font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Olive" title="b1">s</font></span>)</span>,<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Olive" title="b3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Olive" title="b1">s</font> &amp; ( for <font color="Olive" title="b4">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b4">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Olive" title="b1">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b4">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b4">dn</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c17">s</font> be    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c15">n</font>,<font color="Maroon" title="c15">n</font>,<font color="Maroon" title="c16">k</font>, <a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font>, <a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</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="c14">aux</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="scm_comp.html#K21" title="SCM_COMP:func.21">max_Data-Loc_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> implies  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  ex <font color="Olive" title="b2">u</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b3">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b3">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b3">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b3">dn</font></span>)</span> ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E4:36_1"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
<font color="Maroon" title="c14">aux</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="scm_comp.html#K21" title="SCM_COMP:func.21">max_Data-Loc_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  ex <font color="Olive" title="b2">u</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b3">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b3">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b3">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b3">dn</font></span>)</span> ) )</span><br/>

<a NAME="E5:36_1"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
 <a href="scm_comp.html#K21" title="SCM_COMP:func.21">max_Data-Loc_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K21" title="SCM_COMP:func.21">max_Data-Loc_in</a> <font color="Maroon" title="c12">tl</font></span>)</span>,<span class="p1">(<span class="default"><a href="scm_comp.html#K21" title="SCM_COMP:func.21">max_Data-Loc_in</a> <font color="Maroon" title="c13">tr</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scm_comp.html#T12" target="_self" title="SCM_COMP:th.12">Th12</a></span>;<br/>
<a NAME="E6:36_1"/><span class="kw">then </span>
 <a href="scm_comp.html#K21" title="SCM_COMP:func.21">max_Data-Loc_in</a> <font color="Maroon" title="c12">tl</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scm_comp.html#K21" title="SCM_COMP:func.21">max_Data-Loc_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E7:36_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A7</font></span>: 
 <a href="scm_comp.html#K21" title="SCM_COMP:func.21">max_Data-Loc_in</a> <font color="Maroon" title="c12">tl</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E8:36_1"/>
 <a href="scm_comp.html#K21" title="SCM_COMP:func.21">max_Data-Loc_in</a> <font color="Maroon" title="c13">tr</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scm_comp.html#K21" title="SCM_COMP:func.21">max_Data-Loc_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></span>;<br/>
<a NAME="E9:36_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A8</font></span>: 
 <a href="scm_comp.html#K21" title="SCM_COMP:func.21">max_Data-Loc_in</a> <font color="Maroon" title="c13">tr</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E10:36_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A9</font></span>: 
 <a href="scm_comp.html#K21" title="SCM_COMP:func.21">max_Data-Loc_in</a> <font color="Maroon" title="c13">tr</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E11:36_1"/><span class="lab"><font color="Green" title="E36">A10</font></span>: 
 <a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <font color="Maroon" title="c13">tr</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c11">nt</font>,<font color="Maroon" title="c14">aux</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="scm_comp.html#T10" target="_self" title="SCM_COMP:th.10">Th10</a></span>;<br/>
<a NAME="E12:36_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A11</font></span>: 
 <a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <font color="Maroon" title="c13">tr</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c11">nt</font>,<font color="Maroon" title="c14">aux</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T30" title="AFINSQ_1:th.30">AFINSQ_1:30</a></span>;<br/>
<a NAME="E13:36_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">s</font> is    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c15">n</font>,<font color="Maroon" title="c15">n</font>,<font color="Maroon" title="c16">k</font>, <a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c14">aux</font>, <a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scm_comp.html#T1" target="_self" title="SCM_COMP:th.1">Th1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">i1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> , <font color="Maroon" title="c19">u1</font> being   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <span class="kw"> such that </span><br/><a NAME="E15:36_1"/><span class="lab"><font color="Green" title="E38">A12</font></span>: 
<font color="Maroon" title="c19">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E16:36_1"/><span class="lab"><font color="Green" title="E39">A13</font></span>: 
<font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c14">aux</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E17:36_1"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c18">i1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i1</font>
 <span class="kw">and </span><br/><a NAME="E18:36_1"/><span class="lab"><font color="Green" title="E40">A14</font></span>: 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c19">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E19:36_1"/><span class="lab"><font color="Green" title="E41">A15</font></span>: 
<font color="Maroon" title="c19">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font>
 <span class="kw">and </span><br/><a NAME="E20:36_1"/><span class="lab"><font color="Green" title="E42">A16</font></span>: 
for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:36_1"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E7:36_1"><span class="lab"><font color="Green" title="E33">A7</font></span></a></span>;<br/>
<a NAME="E21:36_1"/>
<font color="Maroon" title="c19">u1</font> is    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c16">k</font>,<span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <font color="Maroon" title="c13">tr</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c11">nt</font>,<font color="Maroon" title="c14">aux</font></span>)</span>, <a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:36_1"><span class="lab"><font color="Green" title="E37">A11</font></span></a>, <a class="txt" href="#E15:36_1"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="txt" href="#E16:36_1"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="txt" href="#E18:36_1"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <a class="ref" href="scm_comp.html#T2" target="_self" title="SCM_COMP:th.2">Th2</a></span>;<br/>
<a NAME="E22:36_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">u1</font> is    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c16">k</font>, <a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <font color="Maroon" title="c13">tr</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scm_comp.html#T1" target="_self" title="SCM_COMP:th.1">Th1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c20">i2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> , <font color="Maroon" title="c21">u2</font> being   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <span class="kw"> such that </span><br/><a NAME="E24:36_1"/><span class="lab"><font color="Green" title="E43">A17</font></span>: 
<font color="Maroon" title="c21">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c19">u1</font></span>)</span>,<font color="Maroon" title="c19">u1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E25:36_1"/><span class="lab"><font color="Green" title="E44">A18</font></span>: 
<font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <font color="Maroon" title="c13">tr</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E26:36_1"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c19">u1</font></span>)</span>,<font color="Maroon" title="c19">u1</font>,<font color="Maroon" title="c20">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">i2</font>
 <span class="kw">and </span><br/><a NAME="E27:36_1"/><span class="lab"><font color="Green" title="E45">A19</font></span>: 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c21">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E28:36_1"/><span class="lab"><font color="Green" title="E46">A20</font></span>: 
<font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">u1</font>
 <span class="kw">and </span><br/><a NAME="E29:36_1"/><span class="lab"><font color="Green" title="E47">A21</font></span>: 
for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><font color="Maroon" title="c19">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:36_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E10:36_1"><span class="lab"><font color="Green" title="E35">A9</font></span></a></span>;<br/>
<a NAME="E30:36_1"/><span class="lab"><font color="Green" title="E48">S</font></span>: 
 <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c19">u1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:36_1"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="ref" href="ami_1.html#T123" title="AMI_1:th.123">AMI_1:123</a></span>;<br/>
<a NAME="E31:36_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A22</font></span>: 
<font color="Maroon" title="c21">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:36_1"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="txt" href="#E24:36_1"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E50">A23</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</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="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scm_comp.html#K21" title="SCM_COMP:func.21">max_Data-Loc_in</a> <font color="Maroon" title="c13">tr</font> implies <font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c22">n</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E1:36_1_1"/>
<font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scm_comp.html#K21" title="SCM_COMP:func.21">max_Data-Loc_in</a> <font color="Maroon" title="c13">tr</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c22">n</font></span>)</span></span><br/><a NAME="E2:36_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:36_1"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:36_1_1"/>
<font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c22">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c22">n</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36_1"><span class="lab"><font color="Green" title="E42">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E33:36_1"/><span class="lab"><font color="Green" title="E51">A24</font></span>: 
<font color="Maroon" title="c14">aux</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E34:36_1"/><span class="lab"><font color="Green" title="E52">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K10" title="SCM_COMP:func.10">-Meaning_on</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scm_comp.html#T7" target="_self" title="SCM_COMP:th.7">Th7</a></span>;<br/>
<a NAME="E35:36_1"/><span class="lab"><font color="Green" title="E53">A26</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <span class="p2">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <font color="Maroon" title="c13">tr</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i2</font> <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="#E16:36_1"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="txt" href="#E25:36_1"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:36_1_2"/>
( <font color="Maroon" title="c11">nt</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="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c11">nt</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="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c11">nt</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="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c11">nt</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="numbers.html#K6" title="NUMBERS:func.6">0</a> ,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c11">nt</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="numbers.html#K6" title="NUMBERS:func.6">0</a> ,4</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="scm_comp.html#T3" target="_self" title="SCM_COMP:th.3">Th3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:36_1_2_1"/><span class="lab"><font color="Green" title="E54">A27</font></span>: 
<font color="Maroon" title="c11">nt</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="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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">  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  ex <font color="Olive" title="b2">u</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b3">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b3">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b3">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b3">dn</font></span>)</span> ) )</span><br/><div class="add"><a NAME="E2:36_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A28</font></span>: 
<font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K10" title="SCM_COMP:func.10">-Meaning_on</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scm_comp.html#D8" target="_self" title="SCM_COMP:def.8">Def8</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c22">i</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b2">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b2">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b2">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b2">dn</font></span>)</span> ) )</span><br/><span class="kw">take </span><font color="Maroon" title="c23">u</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><span class="kw">thus </span><a NAME="E3:36_1_2_1"/>
<font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><a NAME="E4:36_1_2_1"/><span class="lab"><font color="Green" title="E56">A29</font></span>: 
 <a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c11">nt</font>,<font color="Maroon" title="c14">aux</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K4" title="AMI_3:func.4">AddTo</a> <span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span>,<span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1_2_1"><span class="lab"><font color="Green" title="E54">A27</font></span></a>, <a class="ref" href="scm_comp.html#D10" target="_self" title="SCM_COMP:def.10">Def10</a></span>;<br/><a NAME="E5:36_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A30</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c11">nt</font>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></span>;<br/><span class="kw">hence </span><a NAME="E6:36_1_2_1"/>
<font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E35:36_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><span class="kw">thus </span><a NAME="E7:36_1_2_1"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:36_1"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="txt" href="#E24:36_1"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="txt" href="#E27:36_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E30:36_1"><span class="lab"><font color="Green" title="E48">S</font></span></a>, <a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><a NAME="E8:36_1_2_1"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E35:36_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E5:36_1_2_1"><span class="lab"><font color="Green" title="E57">A30</font></span></a>, <a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a></span>;<br/><a NAME="E9:36_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A31</font></span>: <font color="Maroon" title="c17">s</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></span>
<br/>.= 
 <a href="ami_3.html#K4" title="AMI_3:func.4">AddTo</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:36_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E11:36_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E4:36_1_2_1"><span class="lab"><font color="Green" title="E56">A29</font></span></a>, <a class="ref" href="afinsq_1.html#T40" title="AFINSQ_1:th.40">AFINSQ_1:40</a></span>
;<br/><span class="kw">hence </span> <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:36_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E31:36_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="ref" href="scm_1.html#T19" title="SCM_1:th.19">SCM_1:19</a></span>
<br/>.= 
<font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

;<br/> <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="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><span class="kw">thus </span><font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:36_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E31:36_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E10:36_1_2_1"><span class="lab"><font color="Green" title="E58">A31</font></span></a>, <a class="ref" href="scm_1.html#T19" title="SCM_1:th.19">SCM_1:19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">u1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:36_1"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="txt" href="#E29:36_1"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="txt" href="#E33:36_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:36_1"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E34:36_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E32:36_1"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="txt" href="#E2:36_1_2_1"><span class="lab"><font color="Green" title="E55">A28</font></span></a>, <a class="ref" href="scm_comp.html#T13" target="_self" title="SCM_COMP:th.13">Th13</a></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">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c24">dn</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</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="c24">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> implies <font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E13:36_1_2_1"/><span class="lab"><font color="Green" title="E59">A32</font></span>: 
<font color="Maroon" title="c24">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span></span><br/><a NAME="E14:36_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c24">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E15:36_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A33</font></span>: 
<font color="Maroon" title="c19">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:36_1"><span class="lab"><font color="Green" title="E47">A21</font></span></a></span>;<br/><a NAME="E16:36_1_2_1"/>
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:36_1_2_1"><span class="lab"><font color="Green" title="E59">A32</font></span></a>, <a class="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></span>;<br/><a NAME="E17:36_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:36_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E31:36_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E10:36_1_2_1"><span class="lab"><font color="Green" title="E58">A31</font></span></a>, <a class="ref" href="scm_1.html#T19" title="SCM_1:th.19">SCM_1:19</a></span>;<br/><span class="kw">hence </span><a NAME="E18:36_1_2_1"/>
<font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36_1"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="txt" href="#E13:36_1_2_1"><span class="lab"><font color="Green" title="E59">A32</font></span></a>, <a class="txt" href="#E15:36_1_2_1"><span class="lab"><font color="Green" title="E60">A33</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:36_1_2_2"/><span class="lab"><font color="Green" title="E54">A34</font></span>: 
<font color="Maroon" title="c11">nt</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="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</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">  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  ex <font color="Olive" title="b2">u</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b3">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b3">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b3">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b3">dn</font></span>)</span> ) )</span><br/><div class="add"><a NAME="E2:36_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A35</font></span>: 
<font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K10" title="SCM_COMP:func.10">-Meaning_on</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scm_comp.html#D8" target="_self" title="SCM_COMP:def.8">Def8</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c22">i</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b2">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b2">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b2">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b2">dn</font></span>)</span> ) )</span><br/><span class="kw">take </span><font color="Maroon" title="c23">u</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><span class="kw">thus </span><a NAME="E3:36_1_2_2"/>
<font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><a NAME="E4:36_1_2_2"/><span class="lab"><font color="Green" title="E56">A36</font></span>: 
 <a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c11">nt</font>,<font color="Maroon" title="c14">aux</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K5" title="AMI_3:func.5">SubFrom</a> <span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span>,<span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1_2_2"><span class="lab"><font color="Green" title="E54">A34</font></span></a>, <a class="ref" href="scm_comp.html#D10" target="_self" title="SCM_COMP:def.10">Def10</a></span>;<br/><a NAME="E5:36_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A37</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c11">nt</font>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></span>;<br/><span class="kw">hence </span><a NAME="E6:36_1_2_2"/>
<font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E35:36_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><span class="kw">thus </span><a NAME="E7:36_1_2_2"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:36_1"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="txt" href="#E24:36_1"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="txt" href="#E27:36_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E30:36_1"><span class="lab"><font color="Green" title="E48">S</font></span></a>, <a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><a NAME="E8:36_1_2_2"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E35:36_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E5:36_1_2_2"><span class="lab"><font color="Green" title="E57">A37</font></span></a>, <a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a></span>;<br/><a NAME="E9:36_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A38</font></span>: <font color="Maroon" title="c17">s</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></span>
<br/>.= 
 <a href="ami_3.html#K5" title="AMI_3:func.5">SubFrom</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E35:36_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E4:36_1_2_2"><span class="lab"><font color="Green" title="E56">A36</font></span></a>, <a class="ref" href="afinsq_1.html#T40" title="AFINSQ_1:th.40">AFINSQ_1:40</a></span>
;<br/><span class="kw">hence </span> <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:36_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E31:36_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="ref" href="scm_1.html#T20" title="SCM_1:th.20">SCM_1:20</a></span>
<br/>.= 
<font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

;<br/> <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="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><span class="kw">thus </span><font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:36_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E31:36_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E10:36_1_2_2"><span class="lab"><font color="Green" title="E58">A38</font></span></a>, <a class="ref" href="scm_1.html#T20" title="SCM_1:th.20">SCM_1:20</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">u1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:36_1"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="txt" href="#E29:36_1"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="txt" href="#E33:36_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:36_1"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E34:36_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E32:36_1"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="txt" href="#E2:36_1_2_2"><span class="lab"><font color="Green" title="E55">A35</font></span></a>, <a class="ref" href="scm_comp.html#T13" target="_self" title="SCM_COMP:th.13">Th13</a></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">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c24">dn</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</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="c24">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> implies <font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E13:36_1_2_2"/><span class="lab"><font color="Green" title="E59">A39</font></span>: 
<font color="Maroon" title="c24">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span></span><br/><a NAME="E14:36_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c24">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E15:36_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A40</font></span>: 
<font color="Maroon" title="c19">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:36_1"><span class="lab"><font color="Green" title="E47">A21</font></span></a></span>;<br/><a NAME="E16:36_1_2_2"/>
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:36_1_2_2"><span class="lab"><font color="Green" title="E59">A39</font></span></a>, <a class="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></span>;<br/><a NAME="E17:36_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:36_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E31:36_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E10:36_1_2_2"><span class="lab"><font color="Green" title="E58">A38</font></span></a>, <a class="ref" href="scm_1.html#T20" title="SCM_1:th.20">SCM_1:20</a></span>;<br/><span class="kw">hence </span><a NAME="E18:36_1_2_2"/>
<font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36_1"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="txt" href="#E13:36_1_2_2"><span class="lab"><font color="Green" title="E59">A39</font></span></a>, <a class="txt" href="#E15:36_1_2_2"><span class="lab"><font color="Green" title="E60">A40</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2_3"><span class="kw">suppose </span></a><a NAME="E1:36_1_2_3"/><span class="lab"><font color="Green" title="E54">A41</font></span>: 
<font color="Maroon" title="c11">nt</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="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</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">  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  ex <font color="Olive" title="b2">u</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b3">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b3">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b3">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b3">dn</font></span>)</span> ) )</span><br/><div class="add"><a NAME="E2:36_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A42</font></span>: 
<font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K10" title="SCM_COMP:func.10">-Meaning_on</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scm_comp.html#D8" target="_self" title="SCM_COMP:def.8">Def8</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c22">i</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b2">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b2">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b2">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b2">dn</font></span>)</span> ) )</span><br/><span class="kw">take </span><font color="Maroon" title="c23">u</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><span class="kw">thus </span><a NAME="E3:36_1_2_3"/>
<font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><a NAME="E4:36_1_2_3"/><span class="lab"><font color="Green" title="E56">A43</font></span>: 
 <a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c11">nt</font>,<font color="Maroon" title="c14">aux</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K6" title="AMI_3:func.6">MultBy</a> <span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span>,<span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1_2_3"><span class="lab"><font color="Green" title="E54">A41</font></span></a>, <a class="ref" href="scm_comp.html#D10" target="_self" title="SCM_COMP:def.10">Def10</a></span>;<br/><a NAME="E5:36_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A44</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c11">nt</font>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></span>;<br/><span class="kw">hence </span><a NAME="E6:36_1_2_3"/>
<font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E35:36_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><span class="kw">thus </span><a NAME="E7:36_1_2_3"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:36_1"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="txt" href="#E24:36_1"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="txt" href="#E27:36_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E30:36_1"><span class="lab"><font color="Green" title="E48">S</font></span></a>, <a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><a NAME="E8:36_1_2_3"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E35:36_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E5:36_1_2_3"><span class="lab"><font color="Green" title="E57">A44</font></span></a>, <a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a></span>;<br/><a NAME="E9:36_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A45</font></span>: <font color="Maroon" title="c17">s</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></span>
<br/>.= 
 <a href="ami_3.html#K6" title="AMI_3:func.6">MultBy</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E35:36_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E4:36_1_2_3"><span class="lab"><font color="Green" title="E56">A43</font></span></a>, <a class="ref" href="afinsq_1.html#T40" title="AFINSQ_1:th.40">AFINSQ_1:40</a></span>
;<br/><span class="kw">hence </span> <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:36_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E31:36_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="ref" href="scm_1.html#T21" title="SCM_1:th.21">SCM_1:21</a></span>
<br/>.= 
<font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

;<br/> <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="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><span class="kw">thus </span><font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:36_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E31:36_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E10:36_1_2_3"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="ref" href="scm_1.html#T21" title="SCM_1:th.21">SCM_1:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">u1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:36_1"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="txt" href="#E29:36_1"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="txt" href="#E33:36_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:36_1"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E34:36_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E32:36_1"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="txt" href="#E2:36_1_2_3"><span class="lab"><font color="Green" title="E55">A42</font></span></a>, <a class="ref" href="scm_comp.html#T13" target="_self" title="SCM_COMP:th.13">Th13</a></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">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c24">dn</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</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="c24">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> implies <font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E13:36_1_2_3"/><span class="lab"><font color="Green" title="E59">A46</font></span>: 
<font color="Maroon" title="c24">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span></span><br/><a NAME="E14:36_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c24">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E15:36_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A47</font></span>: 
<font color="Maroon" title="c19">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:36_1"><span class="lab"><font color="Green" title="E47">A21</font></span></a></span>;<br/><a NAME="E16:36_1_2_3"/>
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:36_1_2_3"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></span>;<br/><a NAME="E17:36_1_2_3"/><span class="kw">then </span>
<font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:36_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E31:36_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E10:36_1_2_3"><span class="lab"><font color="Green" title="E58">A45</font></span></a>, <a class="ref" href="scm_1.html#T21" title="SCM_1:th.21">SCM_1:21</a></span>;<br/><span class="kw">hence </span><a NAME="E18:36_1_2_3"/>
<font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36_1"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="txt" href="#E13:36_1_2_3"><span class="lab"><font color="Green" title="E59">A46</font></span></a>, <a class="txt" href="#E15:36_1_2_3"><span class="lab"><font color="Green" title="E60">A47</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2_4"><span class="kw">suppose </span></a><a NAME="E1:36_1_2_4"/><span class="lab"><font color="Green" title="E54">A48</font></span>: 
<font color="Maroon" title="c11">nt</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="numbers.html#K6" title="NUMBERS:func.6">0</a> ,3</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">  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  ex <font color="Olive" title="b2">u</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b3">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b3">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b3">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b3">dn</font></span>)</span> ) )</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c22">i</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b2">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b2">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b2">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b2">dn</font></span>)</span> ) )</span><br/><span class="kw">take </span><font color="Maroon" title="c23">u</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><span class="kw">thus </span><a NAME="E2:36_1_2_4"/>
<font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><a NAME="E3:36_1_2_4"/><span class="lab"><font color="Green" title="E55">A49</font></span>: 
 <a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c11">nt</font>,<font color="Maroon" title="c14">aux</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K7" title="AMI_3:func.7">Divide</a> <span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span>,<span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1_2_4"><span class="lab"><font color="Green" title="E54">A48</font></span></a>, <a class="ref" href="scm_comp.html#D10" target="_self" title="SCM_COMP:def.10">Def10</a></span>;<br/><a NAME="E4:36_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A50</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c11">nt</font>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></span>;<br/><span class="kw">hence </span><a NAME="E5:36_1_2_4"/>
<font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E35:36_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><a NAME="E6:36_1_2_4"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E35:36_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E4:36_1_2_4"><span class="lab"><font color="Green" title="E56">A50</font></span></a>, <a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a></span>;<br/><a NAME="E7:36_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c22">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A51</font></span>: <font color="Maroon" title="c17">s</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></span>
<br/>.= 
 <a href="ami_3.html#K7" title="AMI_3:func.7">Divide</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E35:36_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E3:36_1_2_4"><span class="lab"><font color="Green" title="E55">A49</font></span></a>, <a class="ref" href="afinsq_1.html#T40" title="AFINSQ_1:th.40">AFINSQ_1:40</a></span>
;<br/><span class="kw">thus </span><a NAME="E9:36_1_2_4"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c22">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:36_1"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="txt" href="#E24:36_1"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="txt" href="#E27:36_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E30:36_1"><span class="lab"><font color="Green" title="E48">S</font></span></a>, <a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><a NAME="E10:36_1_2_4"/><span class="lab"><font color="Green" title="E58">A52</font></span>: 
<font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K10" title="SCM_COMP:func.10">-Meaning_on</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1_2_4"><span class="lab"><font color="Green" title="E54">A48</font></span></a>, <a class="ref" href="scm_comp.html#D8" target="_self" title="SCM_COMP:def.8">Def8</a></span>;<br/><a NAME="E11:36_1_2_4"/>
<font color="Maroon" title="c14">aux</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E12:36_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A53</font></span>: 
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">aux</font> <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="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></span>;<br/><span class="kw">hence </span> <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c23">u</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:36_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E31:36_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E8:36_1_2_4"><span class="lab"><font color="Green" title="E57">A51</font></span></a>, <a class="ref" href="scm_1.html#T22" title="SCM_1:th.22">SCM_1:22</a></span>
<br/>.= 
<font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

;<br/> <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="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><span class="kw">thus </span><font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:36_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E31:36_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E8:36_1_2_4"><span class="lab"><font color="Green" title="E57">A51</font></span></a>, <a class="txt" href="#E12:36_1_2_4"><span class="lab"><font color="Green" title="E59">A53</font></span></a>, <a class="ref" href="scm_1.html#T22" title="SCM_1:th.22">SCM_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">u1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:36_1"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="txt" href="#E29:36_1"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="txt" href="#E33:36_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:36_1"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E34:36_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E32:36_1"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="txt" href="#E10:36_1_2_4"><span class="lab"><font color="Green" title="E58">A52</font></span></a>, <a class="ref" href="scm_comp.html#T13" target="_self" title="SCM_COMP:th.13">Th13</a></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">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c24">dn</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</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="c24">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> implies <font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E15:36_1_2_4"/><span class="lab"><font color="Green" title="E60">A54</font></span>: 
<font color="Maroon" title="c24">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span></span><br/><a NAME="E16:36_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A55</font></span>: 
<font color="Maroon" title="c24">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E17:36_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A56</font></span>: 
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">aux</font> <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="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></span>;<br/><a NAME="E18:36_1_2_4"/><span class="lab"><font color="Green" title="E63">A57</font></span>: 
<font color="Maroon" title="c19">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:36_1"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="txt" href="#E16:36_1_2_4"><span class="lab"><font color="Green" title="E61">A55</font></span></a></span>;<br/><a NAME="E19:36_1_2_4"/>
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:36_1_2_4"><span class="lab"><font color="Green" title="E60">A54</font></span></a>, <a class="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></span>;<br/><a NAME="E20:36_1_2_4"/><span class="kw">then </span>
<font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:36_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E31:36_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E8:36_1_2_4"><span class="lab"><font color="Green" title="E57">A51</font></span></a>, <a class="txt" href="#E12:36_1_2_4"><span class="lab"><font color="Green" title="E59">A53</font></span></a>, <a class="txt" href="#E17:36_1_2_4"><span class="lab"><font color="Green" title="E62">A56</font></span></a>, <a class="ref" href="scm_1.html#T22" title="SCM_1:th.22">SCM_1:22</a></span>;<br/><span class="kw">hence </span><a NAME="E21:36_1_2_4"/>
<font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c24">dn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36_1"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="txt" href="#E15:36_1_2_4"><span class="lab"><font color="Green" title="E60">A54</font></span></a>, <a class="txt" href="#E18:36_1_2_4"><span class="lab"><font color="Green" title="E63">A57</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="36_1_2_5"><span class="kw">suppose </span></a><a NAME="E1:36_1_2_5"/><span class="lab"><font color="Green" title="E54">A58</font></span>: 
<font color="Maroon" title="c11">nt</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="numbers.html#K6" title="NUMBERS:func.6">0</a> ,4</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">  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  ex <font color="Olive" title="b2">u</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b3">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b3">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b3">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b3">dn</font></span>)</span> ) )</span><br/><div class="add"><a NAME="E2:36_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A59</font></span>: 
<font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K10" title="SCM_COMP:func.10">-Meaning_on</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scm_comp.html#D8" target="_self" title="SCM_COMP:def.8">Def8</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c22">i</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c23">u</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><span class="kw">take </span><font color="Maroon" title="c24">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c24">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c24">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">k</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b2">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b2">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b2">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b2">dn</font></span>)</span> ) )</span><br/><span class="kw">take </span><font color="Maroon" title="c25">v</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c25">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c24">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c24">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">k</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c25">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c25">v</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">v</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><span class="kw">thus </span><a NAME="E3:36_1_2_5"/>
<font color="Maroon" title="c25">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c24">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c24">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">k</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c25">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c25">v</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">v</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><a NAME="E4:36_1_2_5"/><span class="lab"><font color="Green" title="E56">A61</font></span>: 
 <a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c11">nt</font>,<font color="Maroon" title="c14">aux</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K7" title="AMI_3:func.7">Divide</a> <span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span>,<span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="afinsq_1.html#K6" title="AFINSQ_1:func.6">%&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:36_1_2_5"><span class="lab"><font color="Green" title="E54">A58</font></span></a>, <a class="ref" href="scm_comp.html#D10" target="_self" title="SCM_COMP:def.10">Def10</a></span>;<br/><a NAME="E5:36_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A62</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c11">nt</font>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T42" title="AFINSQ_1:th.42">AFINSQ_1:42</a></span>;<br/><a NAME="E6:36_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A60</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c11">nt</font>,<font color="Maroon" title="c14">aux</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T88" title="CARD_1:th.88">CARD_1:88</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E7:36_1_2_5"/><span class="lab"><font color="Green" title="E59">A64</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <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="#E11:36_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E35:36_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E5:36_1_2_5"><span class="lab"><font color="Green" title="E57">A62</font></span></a>, <a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a></span>;<br/><span class="kw">hence </span><a NAME="E8:36_1_2_5"/>
<font color="Maroon" title="c24">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c24">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">k</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c25">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c25">v</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">v</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><a NAME="E9:36_1_2_5"/><span class="lab"><font color="Green" title="E60">A65</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c11">nt</font>,<font color="Maroon" title="c14">aux</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:36_1_2_5"><span class="lab"><font color="Green" title="E57">A62</font></span></a>, <a class="ref" href="card_1.html#T88" title="CARD_1:th.88">CARD_1:88</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E10:36_1_2_5"/>
<font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:36_1_2_5"><span class="lab"><font color="Green" title="E59">A64</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A66</font></span>: <font color="Maroon" title="c17">s</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">k</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c24">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c11">nt</font>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E35:36_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E9:36_1_2_5"><span class="lab"><font color="Green" title="E60">A65</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1_2_5"><span class="lab"><font color="Green" title="E56">A61</font></span></a>, <a class="ref" href="afinsq_1.html#T42" title="AFINSQ_1:th.42">AFINSQ_1:42</a></span>
;<br/><a NAME="E12:36_1_2_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E13:36_1_2_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:36_1_2_5"><span class="lab"><font color="Green" title="E59">A64</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A67</font></span>: <font color="Maroon" title="c17">s</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c11">nt</font>,<font color="Maroon" title="c14">aux</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> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E35:36_1"><span class="lab"><font color="Green" title="E53">A26</font></span></a>, <a class="txt" href="#E6:36_1_2_5"><span class="lab"><font color="Green" title="E58">A60</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</a></span>
<br/>.= 
 <a href="ami_3.html#K7" title="AMI_3:func.7">Divide</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_1_2_5"><span class="lab"><font color="Green" title="E56">A61</font></span></a>, <a class="ref" href="afinsq_1.html#T42" title="AFINSQ_1:th.42">AFINSQ_1:42</a></span>
;<br/><a NAME="E15:36_1_2_5"/>
<font color="Maroon" title="c14">aux</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E16:36_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A68</font></span>: 
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">aux</font> <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="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></span>;<br/><span class="kw">hence </span><span class="lab"><font color="Green" title="E64">A69</font></span>:  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c24">k</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:36_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E31:36_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E14:36_1_2_5"><span class="lab"><font color="Green" title="E62">A67</font></span></a>, <a class="ref" href="scm_1.html#T22" title="SCM_1:th.22">SCM_1:22</a></span>
<br/>.= 
<font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">k</font>

;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c25">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c25">v</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">v</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><span class="kw">hence </span> <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c25">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1_2_5"><span class="lab"><font color="Green" title="E61">A66</font></span></a>, <a class="ref" href="scm_1.html#T18" title="SCM_1:th.18">SCM_1:18</a></span>
<br/>.= 
<font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>

;<br/> <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="c25">v</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">v</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/><span class="kw">thus </span><font color="Maroon" title="c25">v</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1_2_5"><span class="lab"><font color="Green" title="E61">A66</font></span></a>, <a class="txt" href="#E17:36_1_2_5"><span class="lab"><font color="Green" title="E64">A69</font></span></a>, <a class="ref" href="scm_1.html#T18" title="SCM_1:th.18">SCM_1:18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:36_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E31:36_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E14:36_1_2_5"><span class="lab"><font color="Green" title="E62">A67</font></span></a>, <a class="txt" href="#E16:36_1_2_5"><span class="lab"><font color="Green" title="E63">A68</font></span></a>, <a class="ref" href="scm_1.html#T22" title="SCM_1:th.22">SCM_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">u1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:36_1"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="txt" href="#E29:36_1"><span class="lab"><font color="Green" title="E47">A21</font></span></a>, <a class="txt" href="#E33:36_1"><span class="lab"><font color="Green" title="E51">A24</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c12">tl</font>,<font color="Maroon" title="c13">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:36_1"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E34:36_1"><span class="lab"><font color="Green" title="E52">A25</font></span></a>, <a class="txt" href="#E32:36_1"><span class="lab"><font color="Green" title="E50">A23</font></span></a>, <a class="txt" href="#E2:36_1_2_5"><span class="lab"><font color="Green" title="E55">A59</font></span></a>, <a class="ref" href="scm_comp.html#T13" target="_self" title="SCM_COMP:th.13">Th13</a></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">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> holds <br/><font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">v</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c26">dn</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c26">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> implies <font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c26">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">v</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c26">dn</font></span>)</span> )</span><br/><span class="kw">assume </span><a NAME="E20:36_1_2_5"/><span class="lab"><font color="Green" title="E65">A70</font></span>: 
<font color="Maroon" title="c26">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c26">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">v</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c26">dn</font></span>)</span></span><br/><a NAME="E21:36_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A71</font></span>: 
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c26">dn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></span>;<br/><a NAME="E22:36_1_2_5"/><span class="lab"><font color="Green" title="E67">A72</font></span>: 
<font color="Maroon" title="c26">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36_1_2_5"><span class="lab"><font color="Green" title="E65">A70</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E23:36_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A73</font></span>: 
<font color="Maroon" title="c19">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c26">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c26">dn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:36_1"><span class="lab"><font color="Green" title="E47">A21</font></span></a></span>;<br/><a NAME="E24:36_1_2_5"/>
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c26">dn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">aux</font> <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="#E22:36_1_2_5"><span class="lab"><font color="Green" title="E67">A72</font></span></a>, <a class="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></span>;<br/><a NAME="E25:36_1_2_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c26">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c26">dn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:36_1"><span class="lab"><font color="Green" title="E45">A19</font></span></a>, <a class="txt" href="#E31:36_1"><span class="lab"><font color="Green" title="E49">A22</font></span></a>, <a class="txt" href="#E14:36_1_2_5"><span class="lab"><font color="Green" title="E62">A67</font></span></a>, <a class="txt" href="#E16:36_1_2_5"><span class="lab"><font color="Green" title="E63">A68</font></span></a>, <a class="txt" href="#E21:36_1_2_5"><span class="lab"><font color="Green" title="E66">A71</font></span></a>, <a class="ref" href="scm_1.html#T22" title="SCM_1:th.22">SCM_1:22</a></span>;<br/><a NAME="E26:36_1_2_5"/><span class="kw">then </span>
<font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c26">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c17">s</font></span>)</span>,<font color="Maroon" title="c17">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c18">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c20">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c26">dn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:36_1"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="txt" href="#E20:36_1_2_5"><span class="lab"><font color="Green" title="E65">A70</font></span></a>, <a class="txt" href="#E23:36_1_2_5"><span class="lab"><font color="Green" title="E68">A73</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E27:36_1_2_5"/>
<font color="Maroon" title="c17">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c26">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">v</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c26">dn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:36_1_2_5"><span class="lab"><font color="Green" title="E61">A66</font></span></a>, <a class="txt" href="#E17:36_1_2_5"><span class="lab"><font color="Green" title="E64">A69</font></span></a>, <a class="txt" href="#E21:36_1_2_5"><span class="lab"><font color="Green" title="E66">A71</font></span></a>, <a class="ref" href="scm_1.html#T18" title="SCM_1:th.18">SCM_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E2:36"/><span class="lab"><font color="Green" title="E28">A74</font></span>: 
for <font color="Olive" title="b1">t</font> being  <a href="dtconstr.html#NM2" title="DTCONSTR:NM.2">Terminal</a> of <a href="scm_comp.html#K1" title="SCM_COMP:func.1">SCM-AE</a>  holds  <font color="Maroon">S<sub>2</sub></font>[ <a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Olive" title="b1">t</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">t</font> be   <a href="dtconstr.html#NM2" title="DTCONSTR:NM.2">Terminal</a> of <a href="scm_comp.html#K1" title="SCM_COMP:func.1">SCM-AE</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[ <a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c12">aux</font>, <font color="Maroon" title="c13">n</font>, <font color="Maroon" title="c14">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</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">s</font> being   <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c13">n</font>,<font color="Maroon" title="c13">n</font>,<font color="Maroon" title="c14">k</font>, <a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span>,<font color="Maroon" title="c12">aux</font>, <a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a>   st <font color="Maroon" title="c12">aux</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="scm_comp.html#K21" title="SCM_COMP:func.21">max_Data-Loc_in</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  ex <font color="Olive" title="b3">u</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  st <br/>( <font color="Olive" title="b3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Olive" title="b1">s</font></span>)</span>,<font color="Olive" title="b1">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span>,<font color="Maroon" title="c12">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Olive" title="b1">s</font></span>)</span>,<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b2">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Olive" title="b3">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b3">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c12">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Olive" title="b1">s</font> &amp; ( for <font color="Olive" title="b4">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b4">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">aux</font> holds <br/><font color="Olive" title="b1">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b4">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b4">dn</font></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c15">s</font> be    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c13">n</font>,<font color="Maroon" title="c13">n</font>,<font color="Maroon" title="c14">k</font>, <a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span>,<font color="Maroon" title="c12">aux</font>, <a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</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="c12">aux</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="scm_comp.html#K21" title="SCM_COMP:func.21">max_Data-Loc_in</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span> implies  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  ex <font color="Olive" title="b2">u</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c15">s</font></span>)</span>,<font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span>,<font color="Maroon" title="c12">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c15">s</font></span>)</span>,<font color="Maroon" title="c15">s</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c12">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c15">s</font> &amp; ( for <font color="Olive" title="b3">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b3">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">aux</font> holds <br/><font color="Maroon" title="c15">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b3">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b3">dn</font></span>)</span> ) ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:36_2"/>
<font color="Maroon" title="c12">aux</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="scm_comp.html#K21" title="SCM_COMP:func.21">max_Data-Loc_in</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  ex <font color="Olive" title="b2">u</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c15">s</font></span>)</span>,<font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span>,<font color="Maroon" title="c12">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c15">s</font></span>)</span>,<font color="Maroon" title="c15">s</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b2">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c12">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c15">s</font> &amp; ( for <font color="Olive" title="b3">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b3">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">aux</font> holds <br/><font color="Maroon" title="c15">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b3">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b3">dn</font></span>)</span> ) )</span><br/>

<span class="kw">take </span><font color="Maroon" title="c16">i</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">u</font> being  <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  st <br/>( <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c15">s</font></span>)</span>,<font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span>,<font color="Maroon" title="c12">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c15">s</font></span>)</span>,<font color="Maroon" title="c15">s</font>,<font color="Maroon" title="c16">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Olive" title="b1">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c12">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c15">s</font> &amp; ( for <font color="Olive" title="b2">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b2">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">aux</font> holds <br/><font color="Maroon" title="c15">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b2">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b2">dn</font></span>)</span> ) )</span><br/>

<span class="kw">take </span><font color="Maroon" title="c17">u</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c15">s</font></span>)</span>,<font color="Maroon" title="c15">s</font>,<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c15">s</font></span>)</span>,<font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span>,<font color="Maroon" title="c12">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c15">s</font></span>)</span>,<font color="Maroon" title="c15">s</font>,<font color="Maroon" title="c16">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c17">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c12">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c15">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">aux</font> holds <br/><font color="Maroon" title="c15">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:36_2"/>
<font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c15">s</font></span>)</span>,<font color="Maroon" title="c15">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span>,<font color="Maroon" title="c12">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c15">s</font></span>)</span>,<font color="Maroon" title="c15">s</font>,<font color="Maroon" title="c16">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c17">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c12">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c15">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">aux</font> holds <br/><font color="Maroon" title="c15">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/>

<a NAME="E3:36_2"/><span class="lab"><font color="Green" title="E29">A75</font></span>: 
<span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c12">aux</font></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p3">(<span class="default"><a href="scm_comp.html#K4" title="SCM_COMP:func.4">@</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c12">aux</font></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K4" title="SCM_COMP:func.4">@</a> <font color="Maroon" title="c11">t</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></span>;<br/>
<a NAME="E4:36_2"/><span class="lab"><font color="Green" title="E30">A76</font></span>: 
 <a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span>,<font color="Maroon" title="c12">aux</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c12">aux</font></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p3">(<span class="default"><a href="scm_comp.html#K4" title="SCM_COMP:func.4">@</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scm_comp.html#T9" target="_self" title="SCM_COMP:th.9">Th9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:36_2"/>
<font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K19" title="SCM_COMP:func.19">SCM-Compile</a> <span class="p2">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span>,<font color="Maroon" title="c12">aux</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c15">s</font></span>)</span>,<font color="Maroon" title="c15">s</font>,<font color="Maroon" title="c16">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c17">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c12">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c15">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">aux</font> holds <br/><font color="Maroon" title="c15">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/>

<a NAME="E6:36_2"/><span class="lab"><font color="Green" title="E31">A77</font></span>: 
<font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c15">s</font></span>)</span>,<font color="Maroon" title="c15">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T13" title="AMI_1:th.13">AMI_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:36_2"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Comput</a> <span class="p2">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c15">s</font></span>)</span>,<font color="Maroon" title="c15">s</font>,<font color="Maroon" title="c16">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c17">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c12">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c15">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">aux</font> holds <br/><font color="Maroon" title="c15">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/>

<a NAME="E8:36_2"/>
(  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1"><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c12">aux</font></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p3">(<span class="default"><a href="scm_comp.html#K4" title="SCM_COMP:func.4">@</a> <font color="Maroon" title="c11">t</font></span>)</span></span>)</span></span><a href="afinsq_1.html#K5" title="AFINSQ_1:func.5">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c13">n</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T38" title="AFINSQ_1:th.38">AFINSQ_1:38</a></span>;<br/>
<a NAME="E9:36_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A78</font></span>: 
(  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> &amp; <font color="Maroon" title="c15">s</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <font color="Maroon" title="c13">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c12">aux</font></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K4" title="SCM_COMP:func.4">@</a> <font color="Maroon" title="c11">t</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:36_2"><span class="lab"><font color="Green" title="E30">A76</font></span></a>, <a class="txt" href="#E3:36_2"><span class="lab"><font color="Green" title="E29">A75</font></span></a>, <a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:36_2"/>
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c17">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i</font> <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="#E6:36_2"><span class="lab"><font color="Green" title="E31">A77</font></span></a>, <a class="ref" href="scm_1.html#T18" title="SCM_1:th.18">SCM_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c12">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c15">s</font> &amp; ( for <font color="Olive" title="b1">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">aux</font> holds <br/><font color="Maroon" title="c15">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> ) )</span><br/>

<span class="kw">thus </span><font color="Maroon" title="c17">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c12">aux</font></span>)</span> = 
<font color="Maroon" title="c15">s</font> <a href="scm_comp.html#K11" title="SCM_COMP:func.11">.</a> <font color="Maroon" title="c11">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36_2"><span class="lab"><font color="Green" title="E31">A77</font></span></a>, <a class="txt" href="#E9:36_2"><span class="lab"><font color="Green" title="E32">A78</font></span></a>, <a class="ref" href="scm_1.html#T18" title="SCM_1:th.18">SCM_1:18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c11">t</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c15">s</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="scm_comp.html#T6" target="_self" title="SCM_COMP:th.6">Th6</a></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">dn</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">aux</font> holds <br/><font color="Maroon" title="c15">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Olive" title="b1">dn</font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c18">dn</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</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="c18">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">aux</font> implies <font color="Maroon" title="c15">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c18">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c18">dn</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E12:36_2"/>
<font color="Maroon" title="c18">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">aux</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c18">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c18">dn</font></span>)</span></span><br/>

<a NAME="E13:36_2"/><span class="kw">then </span>
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c18">dn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c12">aux</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:36_2"/>
<font color="Maroon" title="c15">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c18">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">u</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c18">dn</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:36_2"><span class="lab"><font color="Green" title="E31">A77</font></span></a>, <a class="txt" href="#E9:36_2"><span class="lab"><font color="Green" title="E32">A78</font></span></a>, <a class="ref" href="scm_1.html#T18" title="SCM_1:th.18">SCM_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E3:36"/>
for <font color="Olive" title="b1">term</font> being  <a href="scm_comp.html#NM1" title="SCM_COMP:NM.1">bin-term</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">term</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="bintree1.html#S2" title="BINTREE1:sch.2">BINTREE1:sch 2</a>(<a class="txt" href="#E2:36"><span class="lab"><font color="Green" title="E28">A74</font></span></a>, <a class="txt" href="#E1:36"><span class="lab"><font color="Green" title="E27">A1</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
