YES
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<title>Left Termination proof of ../tpdb/LP/talp/maria/hanoiapp.suc.pl</title>
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<BR><B>Left Termination</B> of the query pattern
shanoi_in_5(g, g, g, g, a)
w.r.t. the given <I>Prolog program</I> could successfully be <font color=#00ff00>proven</font>:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>shanoi(s(0), A, B, C, .(mv(A, C), [])).<BR>shanoi(s(s(X)), A, B, C, M)&#160;:-&#160;','(eq(N1, s(X)), ','(shanoi(N1, A, C, B, M1), ','(shanoi(N1, B, A, C, M2), ','(append(M1, .(mv(A, C), []), T), append(T, M2, M))))).<BR>append([], L, L).<BR>append(.(H, L), L1, .(H, R))&#160;:-&#160;append(L, L1, R).<BR>eq(X, X).<BR><BR>Queries:<BR><BR>shanoi(g,g,g,g,a).<BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>shanoi_in</font>: (b,b,b,b,f)
<BR><FONT COLOR=#0000cc>append_in</font>: (b,b,f)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 <B>PiTRS</B></pre><pre>      &#8627 DependencyPairsProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<BR><BR>Using Dependency Pairs [1,30] we result in the following initial DP problem:<BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>EQ_IN_AG</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>EQ_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>EQ_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 <B>PiDP</B></pre><pre>          &#8627 DependencyGraphProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>EQ_IN_AG</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GGGGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>EQ_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>EQ_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GGGGA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GGGGA</font>(<FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 2 SCCs with 7 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>)<BR>The graph contains the following edges 1 > 1, 2 >= 2<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 PiDPToQDPProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x6</font>, <FONT COLOR=#cc0000>x7</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 PiDPToQDPProof</pre><pre>                  &#8627 <B>QDP</B></pre><pre>                    &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))) at position [3] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 PiDPToQDPProof</pre><pre>                  &#8627 QDP</pre><pre>                    &#8627 Rewriting</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPOrderProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We use the reduction pair processor [15].<P><BR>The following pairs can be oriented strictly and are deleted.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)))</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))</BLOCKQUOTE>Used ordering:  Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>.</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>0</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_GGGGA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_gggga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_GGGGA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_gggga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>, x<SUB>5</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_gggga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_gggga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U5_gggga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U6_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>[]</font></B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>append_in_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>append_out_gga</font>(x<SUB>1</SUB>)</B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>eq_in_ag</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>eq_out_ag</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>mv</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>s</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>shanoi_in_gggga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>shanoi_out_gggga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR>The following usable rules [17] were oriented:
none<BR><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 PiDPToQDPProof</pre><pre>                  &#8627 QDP</pre><pre>                    &#8627 Rewriting</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 QDPOrderProof</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 DependencyGraphProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>)
<BR><FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>SHANOI_IN_GGGGA</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>U1_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>0</font>), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>eq_out_ag</font>(<FONT COLOR=#cc0000>N1</font>)) &#8594; <FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>B</font>))
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>N1</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M2</font>)) &#8594; <FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>M1</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>mv</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L</font>), <FONT COLOR=#cc0000>L1</font>) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L1</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>M2</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M2</font>))
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>shanoi_out_gggga</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>shanoi_in_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>eq_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>U2_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U3_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_gggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_gggga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 3 less nodes.<BR><BR></body>

