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

<span class="kw">let </span><font color="Maroon" title="c1">R</font> be   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V7" title="REWRITE1:attr.7">confluent</a>  iff <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  is  <a href="rewrite1.html#V9" title="REWRITE1:attr.9">locally-confluent</a>  )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  is  <a href="rewrite1.html#V9" title="REWRITE1:attr.9">locally-confluent</a>  implies <font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V7" title="REWRITE1:attr.7">confluent</a>  )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:84_1"/><span class="lab"><font color="Green" title="E67">A1</font></span>: 
<font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V7" title="REWRITE1:attr.7">confluent</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  is  <a href="rewrite1.html#V9" title="REWRITE1:attr.9">locally-confluent</a> </span><br/><span class="kw">thus </span><a NAME="E2:84_1"/>
<font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  is  <a href="rewrite1.html#V9" title="REWRITE1:attr.9">locally-confluent</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rewrite1.html#D24" target="_self" title="REWRITE1:def.24">REWRITE1:def 24</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  implies <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:84_1_1"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a> </span><br/>

<a NAME="E2:84_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#T20" target="_self" title="REWRITE1:th.20">Th21</a></span>;<br/>
<a NAME="E3:84_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="rewrite1.html#R6" title="REWRITE1:pred.6">are_divergent_wrt</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D8" target="_self" title="REWRITE1:def.8">Def8</a></span>;<br/>
<a NAME="E4:84_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:84_1"><span class="lab"><font color="Green" title="E67">A1</font></span></a>, <a class="ref" href="rewrite1.html#D22" target="_self" title="REWRITE1:def.22">Def22</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:84_1_1"/><span class="lab"><font color="Green" title="E68">A2</font></span>: 
( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">Def7</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">d</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">REWRITE1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> )</span><br/>

<span class="kw">thus </span><a NAME="E7:84_1_1"/>
( <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:84_1_1"><span class="lab"><font color="Green" title="E68">A2</font></span></a>, <a class="ref" href="rewrite1.html#T21" target="_self" title="REWRITE1:th.21">Th22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:84"/><span class="lab"><font color="Green" title="E67">A3</font></span>: 
 for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  holds <br/><font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rewrite1.html#D24" target="_self" title="REWRITE1:def.24">REWRITE1:def 24</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">R</font> is  <a href="rewrite1.html#V7" title="REWRITE1:attr.7">confluent</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rewrite1.html#D22" target="_self" title="REWRITE1:def.22">REWRITE1:def 22</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R6" title="REWRITE1:pred.6">are_divergent_wrt</a> <font color="Maroon" title="c1">R</font> implies <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c4">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><a NAME="E4:84"/><span class="lab"><font color="Green" title="E68">A4</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">and </span><br/><a NAME="E5:84"/><span class="lab"><font color="Green" title="E69">A5</font></span>: 
<font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rewrite1.html#D8" target="_self" title="REWRITE1:def.8">REWRITE1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font></span><br/>

<a NAME="E6:84"/><span class="lab"><font color="Green" title="E70">A6</font></span>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  or <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:84"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="ref" href="rewrite1.html#T20" target="_self" title="REWRITE1:th.20">Th21</a></span>;<br/>
<a NAME="E7:84"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  or <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:84"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="ref" href="rewrite1.html#T20" target="_self" title="REWRITE1:th.20">Th21</a></span>;<br/>
<a NAME="E8:84"/><span class="kw">then </span>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  or <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:84"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="txt" href="#E6:84"><span class="lab"><font color="Green" title="E70">A6</font></span></a>, <a class="ref" href="rewrite1.html#T15" target="_self" title="REWRITE1:th.15">Th16</a>, <a class="ref" href="rewrite1.html#T38" target="_self" title="REWRITE1:th.38">Th39</a></span>;<br/>
<a NAME="E9:84"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="rewrite1.html#R5" title="REWRITE1:pred.5">are_convergent_wrt</a> <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#T36" target="_self" title="REWRITE1:th.36">Th37</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:84"/><span class="lab"><font color="Green" title="E71">A7</font></span>: 
( <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c1">R</font> <a href="finseq_1.html#K18" title="FINSEQ_1:func.18">[*]</a>  <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">Def7</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">d</font>
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="rewrite1.html#D7" target="_self" title="REWRITE1:def.7">REWRITE1:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font> )</span><br/>

<span class="kw">thus </span><a NAME="E12:84"/>
( <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">d</font> &amp; <font color="Maroon" title="c1">R</font> <a href="rewrite1.html#R1" title="REWRITE1:pred.1">reduces</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:84"><span class="lab"><font color="Green" title="E71">A7</font></span></a>, <a class="ref" href="rewrite1.html#T21" target="_self" title="REWRITE1:th.21">Th22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
