YES
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<title>Left Termination proof of ../tpdb/LP/SGST06/btappend.pl</title>
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<BR><B>Left Termination</B> of the query pattern
goal_in_1(g)
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>tappend(nil, T, T).<BR>tappend(node(nil, X, T2), T1, node(T1, X, T2)).<BR>tappend(node(T1, X, nil), T2, node(T1, X, T2)).<BR>tappend(node(T1, X, T2), T3, node(U, X, T2))&#160;:-&#160;tappend(T1, T3, U).<BR>tappend(node(T1, X, T2), T3, node(T1, X, U))&#160;:-&#160;tappend(T2, T3, U).<BR>s2t(s(X), node(T, Y, T))&#160;:-&#160;s2t(X, T).<BR>s2t(s(X), node(nil, Y, T))&#160;:-&#160;s2t(X, T).<BR>s2t(s(X), node(T, Y, nil))&#160;:-&#160;s2t(X, T).<BR>s2t(s(X), node(nil, Y, nil)).<BR>s2t(0, nil).<BR>goal(X)&#160;:-&#160;','(s2t(X, T1), tappend(T1, T2, T3)).<BR><BR>Queries:<BR><BR>goal(g).<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>goal_in</font>: (b)
<BR><FONT COLOR=#0000cc>s2t_in</font>: (b,f)
<BR><FONT COLOR=#0000cc>tappend_in</font>: (b,f,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>goal_in_g</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>)) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>nil</font>), <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>nil</font>), <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>)) &#8594; <FONT COLOR=#0000cc>goal_out_g</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>goal_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>x1</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>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>nil</font>&#160; = &#160;<FONT COLOR=#0000cc>nil</font><BR>
<FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>tappend_out_gaa</font><BR>
<FONT COLOR=#0000cc>U1_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>goal_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_out_g</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>goal_in_g</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>)) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>nil</font>), <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>nil</font>), <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>)) &#8594; <FONT COLOR=#0000cc>goal_out_g</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>goal_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>x1</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>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>nil</font>&#160; = &#160;<FONT COLOR=#0000cc>nil</font><BR>
<FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>tappend_out_gaa</font><BR>
<FONT COLOR=#0000cc>U1_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>goal_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_out_g</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>GOAL_IN_G</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>))
<BR><FONT COLOR=#0000cc>GOAL_IN_G</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>)
<BR><FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>)) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>))
<BR><FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>)) &#8594; <FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>)
<BR><FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)
<BR><FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U2_GAA</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>goal_in_g</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>)) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>nil</font>), <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>nil</font>), <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>)) &#8594; <FONT COLOR=#0000cc>goal_out_g</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>goal_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>x1</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>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>nil</font>&#160; = &#160;<FONT COLOR=#0000cc>nil</font><BR>
<FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>tappend_out_gaa</font><BR>
<FONT COLOR=#0000cc>U1_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>goal_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_out_g</font><BR>
<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_GAA</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>U2_GAA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>GOAL_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>GOAL_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_GAA</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_GAA</font>(<FONT COLOR=#cc0000>x6</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>GOAL_IN_G</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>))
<BR><FONT COLOR=#0000cc>GOAL_IN_G</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>)
<BR><FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>)) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>))
<BR><FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>)) &#8594; <FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>)
<BR><FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)
<BR><FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U2_GAA</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>goal_in_g</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>)) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>nil</font>), <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>nil</font>), <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>)) &#8594; <FONT COLOR=#0000cc>goal_out_g</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>goal_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>x1</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>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>nil</font>&#160; = &#160;<FONT COLOR=#0000cc>nil</font><BR>
<FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>tappend_out_gaa</font><BR>
<FONT COLOR=#0000cc>U1_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>goal_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_out_g</font><BR>
<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_GAA</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>U2_GAA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_G</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>GOAL_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>GOAL_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_GAA</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_GAA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 2 SCCs with 9 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>TAPPEND_IN_GAA</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)
<BR><FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>goal_in_g</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>)) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>nil</font>), <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>nil</font>), <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>)) &#8594; <FONT COLOR=#0000cc>goal_out_g</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>goal_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>x1</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>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>nil</font>&#160; = &#160;<FONT COLOR=#0000cc>nil</font><BR>
<FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>tappend_out_gaa</font><BR>
<FONT COLOR=#0000cc>U1_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>goal_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_out_g</font><BR>
<FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>x1</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>TAPPEND_IN_GAA</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)
<BR><FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>x1</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>TAPPEND_IN_GAA</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>T2</font>)
<BR><FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>T1</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>TAPPEND_IN_GAA</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>T1</font>)<BR>The graph contains the following edges 1 > 1<P></LI>
<LI><FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>TAPPEND_IN_GAA</font>(<FONT COLOR=#cc0000>T2</font>)<BR>The graph contains the following edges 1 > 1<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 UsableRulesProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>goal_in_g</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T1</font>)) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>nil</font>), <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>nil</font>), <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>U</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#cc0000>U</font>)) &#8594; <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#cc0000>T3</font>, <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>T3</font>)) &#8594; <FONT COLOR=#0000cc>goal_out_g</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>goal_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2t_in_ga</font>(<FONT COLOR=#cc0000>x1</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>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2t_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>tappend_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>nil</font>&#160; = &#160;<FONT COLOR=#0000cc>nil</font><BR>
<FONT COLOR=#0000cc>tappend_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>tappend_out_gaa</font><BR>
<FONT COLOR=#0000cc>U1_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>goal_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_out_g</font><BR>
<FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>x1</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 PiDP</pre><pre>                &#8627 UsableRulesProof</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>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>node</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<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>node</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>node</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>nil</font>&#160; = &#160;<FONT COLOR=#0000cc>nil</font><BR>
<FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>x1</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 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>X</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>S2T_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>S2T_IN_GA</font>(<FONT COLOR=#cc0000>X</font>)<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR></body>

