YES
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<title>Left Termination proof of ../tpdb/LP/SGST06/applast.pl</title>
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<BR><B>Left Termination</B> of the query pattern
goal_in_3(g, a, a)
w.r.t. the given <I>Prolog program</I> could successfully be <font color=#00ff00>proven</font>:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>goal(A, B, C)&#160;:-&#160;','(s2l(A, D), applast(D, B, C)).<BR>applast(L, X, Last)&#160;:-&#160;','(append(L, .(X, []), LX), last(Last, LX)).<BR>last(X, .(X, [])).<BR>last(X, .(H, T))&#160;:-&#160;last(X, T).<BR>append([], L, L).<BR>append(.(H, L1), L2, .(H, L3))&#160;:-&#160;append(L1, L2, L3).<BR>s2l(s(X), .(Y, Xs))&#160;:-&#160;s2l(X, Xs).<BR>s2l(0, []).<BR><BR>Queries:<BR><BR>goal(g,a,a).<BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>goal_in</font>: (b,f,f)
<BR><FONT COLOR=#0000cc>s2l_in</font>: (b,f)
<BR><FONT COLOR=#0000cc>applast_in</font>: (b,f,f)
<BR><FONT COLOR=#0000cc>append_in</font>: (b,b,f)
<BR><FONT COLOR=#0000cc>last_in</font>: (f,b)
<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_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#cc0000>LX</font>))
<BR><FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#cc0000>LX</font>)) &#8594; <FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>)
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>goal_out_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>goal_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_in_gaa</font>(<FONT COLOR=#cc0000>x1</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>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2l_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>U7_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>U7_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</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>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_ag</font><BR>
<FONT COLOR=#0000cc>U5_ag</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_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>applast_out_gaa</font><BR>
<FONT COLOR=#0000cc>goal_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_out_gaa</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_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#cc0000>LX</font>))
<BR><FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#cc0000>LX</font>)) &#8594; <FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>)
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>goal_out_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>goal_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_in_gaa</font>(<FONT COLOR=#cc0000>x1</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>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2l_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>U7_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>U7_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</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>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_ag</font><BR>
<FONT COLOR=#0000cc>U5_ag</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_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>applast_out_gaa</font><BR>
<FONT COLOR=#0000cc>goal_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_out_gaa</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_GAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>GOAL_IN_GAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>S2L_IN_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)
<BR><FONT COLOR=#0000cc>S2L_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>S2L_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>S2L_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>U2_GAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>APPLAST_IN_GAA</font>(<FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>APPLAST_IN_GAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>))
<BR><FONT COLOR=#0000cc>APPLAST_IN_GAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>)
<BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>)) &#8594; <FONT COLOR=#0000cc>U4_GAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#cc0000>LX</font>))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_AG</font>(<FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#cc0000>LX</font>)
<BR><FONT COLOR=#0000cc>LAST_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>LAST_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_AG</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_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#cc0000>LX</font>))
<BR><FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#cc0000>LX</font>)) &#8594; <FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>)
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>goal_out_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>goal_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_in_gaa</font>(<FONT COLOR=#cc0000>x1</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>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2l_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>U7_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>U7_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</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>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_ag</font><BR>
<FONT COLOR=#0000cc>U5_ag</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_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>applast_out_gaa</font><BR>
<FONT COLOR=#0000cc>goal_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_out_gaa</font><BR>
<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_GAA</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_GAA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>S2L_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>S2L_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>LAST_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LAST_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_GAA</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_GAA</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>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GAA</font>(<FONT COLOR=#cc0000>x4</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>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>GOAL_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>GOAL_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_AG</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_AG</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_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>U7_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>APPLAST_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPLAST_IN_GAA</font>(<FONT COLOR=#cc0000>x1</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_GAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>GOAL_IN_GAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>S2L_IN_GA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)
<BR><FONT COLOR=#0000cc>S2L_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U7_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>S2L_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>S2L_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>U2_GAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>APPLAST_IN_GAA</font>(<FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>APPLAST_IN_GAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>))
<BR><FONT COLOR=#0000cc>APPLAST_IN_GAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>)
<BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>)) &#8594; <FONT COLOR=#0000cc>U4_GAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#cc0000>LX</font>))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_AG</font>(<FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#cc0000>LX</font>)
<BR><FONT COLOR=#0000cc>LAST_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>LAST_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_AG</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_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#cc0000>LX</font>))
<BR><FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#cc0000>LX</font>)) &#8594; <FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>)
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>goal_out_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>goal_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_in_gaa</font>(<FONT COLOR=#cc0000>x1</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>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2l_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>U7_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>U7_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</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>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_ag</font><BR>
<FONT COLOR=#0000cc>U5_ag</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_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>applast_out_gaa</font><BR>
<FONT COLOR=#0000cc>goal_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_out_gaa</font><BR>
<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_GAA</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_GAA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>S2L_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>S2L_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>LAST_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LAST_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_GAA</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_GAA</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>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GAA</font>(<FONT COLOR=#cc0000>x4</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>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>GOAL_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>GOAL_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_AG</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_AG</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_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>U7_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>APPLAST_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPLAST_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 3 SCCs with 11 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><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LAST_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_AG</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_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#cc0000>LX</font>))
<BR><FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#cc0000>LX</font>)) &#8594; <FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>)
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>goal_out_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>goal_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_in_gaa</font>(<FONT COLOR=#cc0000>x1</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>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2l_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>U7_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>U7_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</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>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_ag</font><BR>
<FONT COLOR=#0000cc>U5_ag</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_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>applast_out_gaa</font><BR>
<FONT COLOR=#0000cc>goal_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_out_gaa</font><BR>
<FONT COLOR=#0000cc>LAST_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LAST_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LAST_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_AG</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>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>LAST_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LAST_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LAST_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_AG</font>(<FONT COLOR=#cc0000>T</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>LAST_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_AG</font>(<FONT COLOR=#cc0000>T</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><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>goal_in_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#cc0000>LX</font>))
<BR><FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#cc0000>LX</font>)) &#8594; <FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>)
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>goal_out_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>goal_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_in_gaa</font>(<FONT COLOR=#cc0000>x1</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>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2l_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>U7_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>U7_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</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>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_ag</font><BR>
<FONT COLOR=#0000cc>U5_ag</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_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>applast_out_gaa</font><BR>
<FONT COLOR=#0000cc>goal_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_out_gaa</font><BR>
<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>) &#8594; <FONT COLOR=#0000cc>APPEND_IN_GGA</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>)<BR>The graph contains the following edges 1 > 1, 2 >= 2<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 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>S2L_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>S2L_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>goal_in_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>))
<BR><FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>U7_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>))
<BR><FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>))
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L1</font>, <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#cc0000>L3</font>)) &#8594; <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L1</font>), <FONT COLOR=#cc0000>L2</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>L3</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>LX</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#cc0000>LX</font>))
<BR><FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>Last</font>, <FONT COLOR=#cc0000>LX</font>)) &#8594; <FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Last</font>)
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)) &#8594; <FONT COLOR=#0000cc>goal_out_gaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>goal_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_in_gaa</font>(<FONT COLOR=#cc0000>x1</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>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>s2l_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2l_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>U7_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>U7_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>s2l_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</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>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>applast_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>append_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>last_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_ag</font><BR>
<FONT COLOR=#0000cc>U5_ag</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_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>applast_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>applast_out_gaa</font><BR>
<FONT COLOR=#0000cc>goal_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>goal_out_gaa</font><BR>
<FONT COLOR=#0000cc>S2L_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>S2L_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 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>S2L_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>S2L_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</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>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>S2L_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>S2L_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 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>S2L_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>S2L_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>S2L_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>S2L_IN_GA</font>(<FONT COLOR=#cc0000>X</font>)<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR></body>

