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<title>Left Termination proof of ../tpdb/LP/talp/apt/curry_ap.pl</title>
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<BR><B>Left Termination</B> of the query pattern
type_in_3(g, g, a)
w.r.t. the given <I>Prolog program</I> could not be shown:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><BR>Clauses:<BR><BR>type(E, var(X), T)&#160;:-&#160;in(E, X, T).<BR>type(E, apply(M, N), T)&#160;:-&#160;','(type(E, M, arrow(S, T)), type(E, N, S)).<BR>type(E, lambda(X, M), arrow(S, T))&#160;:-&#160;type(.(','(X, S), E), M, T).<BR>in(.(','(X, T), E), X, T).<BR>in(.(','(Y, T), E), X, T)&#160;:-&#160;','(\==(X, Y), in(E, X, T)).<BR><BR>Queries:<BR><BR>type(g,g,a).<BR><BR>Added facts for all undefined predicates.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><pre>    &#8627 <B>Prolog</B></pre><pre>      &#8627 PrologToPiTRSProof</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>type(E, var(X), T)&#160;:-&#160;in(E, X, T).<BR>type(E, apply(M, N), T)&#160;:-&#160;','(type(E, M, arrow(S, T)), type(E, N, S)).<BR>type(E, lambda(X, M), arrow(S, T))&#160;:-&#160;type(.(','(X, S), E), M, T).<BR>in(.(','(X, T), E), X, T).<BR>in(.(','(Y, T), E), X, T)&#160;:-&#160;','(\==(X, Y), in(E, X, T)).<BR>\==(X0, X1).<BR><BR>Queries:<BR><BR>type(g,g,a).<BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>type_in</font>: (b,b,f) (f,b,f)
<BR><FONT COLOR=#0000cc>in_in</font>: (b,b,f) (b,b,b) (f,b,f)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</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_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</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>x1</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_gg</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>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_ggg</font><BR>
<FONT COLOR=#0000cc>U5_ggg</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>U5_ggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_ggg</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_ggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_gga</font><BR>
<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_aga</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_aga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_aga</font><BR>
<FONT COLOR=#0000cc>U5_aga</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>U5_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font><BR>
<FONT COLOR=#0000cc>U6_aga</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_aga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_aga</font><BR>
<FONT COLOR=#0000cc>U2_aga</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>U2_aga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_aga</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>U3_aga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_gga</font>(<FONT COLOR=#cc0000>x5</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 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</pre><pre>        &#8627 <B>PiTRS</B></pre><pre>          &#8627 DependencyPairsProof</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</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_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</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>x1</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_gg</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>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_ggg</font><BR>
<FONT COLOR=#0000cc>U5_ggg</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>U5_ggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_ggg</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_ggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_gga</font><BR>
<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_aga</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_aga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_aga</font><BR>
<FONT COLOR=#0000cc>U5_aga</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>U5_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font><BR>
<FONT COLOR=#0000cc>U6_aga</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_aga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_aga</font><BR>
<FONT COLOR=#0000cc>U2_aga</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>U2_aga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_aga</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>U3_aga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_gga</font>(<FONT COLOR=#cc0000>x5</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>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>\==_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U5_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>\==_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>\==_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</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_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</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>x1</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_gg</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>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_ggg</font><BR>
<FONT COLOR=#0000cc>U5_ggg</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>U5_ggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_ggg</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_ggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_gga</font><BR>
<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_aga</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_aga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_aga</font><BR>
<FONT COLOR=#0000cc>U5_aga</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>U5_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font><BR>
<FONT COLOR=#0000cc>U6_aga</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_aga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_aga</font><BR>
<FONT COLOR=#0000cc>U2_aga</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>U2_aga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_aga</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>U3_aga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_GGA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AGA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_AGA</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_AGA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_GGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_AGA</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_AGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>\==_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_IN_GA</font>(<FONT COLOR=#cc0000>x1</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>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_AGA</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>U5_AGA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_GGG</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_GGG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GGA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_GGG</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>U5_GGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_AGA</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>U2_AGA</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_GGA</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_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_AGA</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>U3_AGA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><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><pre>      &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>\==_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U5_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>\==_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>\==_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</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_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</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>x1</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_gg</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>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_ggg</font><BR>
<FONT COLOR=#0000cc>U5_ggg</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>U5_ggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_ggg</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_ggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_gga</font><BR>
<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_aga</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_aga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_aga</font><BR>
<FONT COLOR=#0000cc>U5_aga</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>U5_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font><BR>
<FONT COLOR=#0000cc>U6_aga</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_aga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_aga</font><BR>
<FONT COLOR=#0000cc>U2_aga</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>U2_aga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_aga</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>U3_aga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_GGA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AGA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_AGA</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_AGA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_GGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_AGA</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_AGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>\==_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_IN_GA</font>(<FONT COLOR=#cc0000>x1</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>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_AGA</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>U5_AGA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_GGG</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_GGG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GGA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_GGG</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>U5_GGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_AGA</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>U2_AGA</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_GGA</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_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_AGA</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>U3_AGA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 4 SCCs with 17 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><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><pre>                  &#8627 PiDP</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>E</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>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</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_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</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>x1</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_gg</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>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_ggg</font><BR>
<FONT COLOR=#0000cc>U5_ggg</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>U5_ggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_ggg</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_ggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_gga</font><BR>
<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_aga</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_aga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_aga</font><BR>
<FONT COLOR=#0000cc>U5_aga</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>U5_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font><BR>
<FONT COLOR=#0000cc>U6_aga</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_aga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_aga</font><BR>
<FONT COLOR=#0000cc>U2_aga</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>U2_aga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_aga</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>U3_aga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_AGA</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>U5_AGA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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 UndefinedPredicateHandlerProof</pre><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><pre>                  &#8627 PiDP</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>E</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>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font><BR>
<FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_AGA</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>U5_AGA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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 UndefinedPredicateHandlerProof</pre><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 Rewriting</pre><pre>                  &#8627 PiDP</pre><pre>                  &#8627 PiDP</pre><pre>                  &#8627 PiDP</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>)) at position [1] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><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 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                  &#8627 PiDP</pre><pre>                  &#8627 PiDP</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><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 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 UsableRulesProof</pre><pre>                                  &#8627 <B>QDP</B></pre><pre>                                    &#8627 QReductionProof</pre><pre>                  &#8627 PiDP</pre><pre>                  &#8627 PiDP</pre><pre>                  &#8627 PiDP</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>)</BLOCKQUOTE><BR>R is empty.<BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><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 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 UsableRulesProof</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 QReductionProof</pre><pre>                                      &#8627 <B>QDP</B></pre><pre>                                        &#8627 NonTerminationProof</pre><pre>                  &#8627 PiDP</pre><pre>                  &#8627 PiDP</pre><pre>                  &#8627 PiDP</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>We used the non-termination processor [17] to show that the DP problem is infinite.<BR>Found a loop by narrowing to the left:<BR><BR>The TRS P consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:none<BR><BR><BR>s = <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X'</font>) evaluates to  t =<FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X'</font>)<BR><BR>Thus s starts an infinite chain as s semiunifies with t with the following substitutions:<BR><UL><LI> Semiunifier: [ ]</LI>
<LI> Matcher: [ ]</LI></UL><BR><BR><hr><BR><B>Rewriting sequence</B><BR><BR><B><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X'</font>)</B> &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X'</font>, <FONT COLOR=#0000cc>\==_out_ga</font>)<BR>with rule <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X''</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X''</font>, <FONT COLOR=#0000cc>\==_out_ga</font>) at position [] and matcher [<FONT COLOR=#cc0000>X''</font> / <FONT COLOR=#cc0000>X'</font>]<BR><BR><B><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X'</font>, <FONT COLOR=#0000cc>\==_out_ga</font>)</B> &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X'</font>)<BR>with rule <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>)<BR><BR>Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence<BR><BR><BR>All these steps are and every following step will be a correct step w.r.t to Q.<BR><BR><BR><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><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><pre>                  &#8627 PiDP</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</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_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</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>x1</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_gg</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>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_ggg</font><BR>
<FONT COLOR=#0000cc>U5_ggg</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>U5_ggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_ggg</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_ggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_gga</font><BR>
<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_aga</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_aga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_aga</font><BR>
<FONT COLOR=#0000cc>U5_aga</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>U5_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font><BR>
<FONT COLOR=#0000cc>U6_aga</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_aga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_aga</font><BR>
<FONT COLOR=#0000cc>U2_aga</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>U2_aga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_aga</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>U3_aga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_AGA</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>U2_AGA</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TYPE_IN_AGA</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 UndefinedPredicateHandlerProof</pre><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><pre>                  &#8627 PiDP</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>var</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>x1</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_aga</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_aga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_aga</font><BR>
<FONT COLOR=#0000cc>U5_aga</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>U5_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font><BR>
<FONT COLOR=#0000cc>U6_aga</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_aga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_aga</font><BR>
<FONT COLOR=#0000cc>U2_aga</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>U2_aga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_aga</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>U3_aga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_AGA</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>U2_AGA</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TYPE_IN_AGA</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 UndefinedPredicateHandlerProof</pre><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><pre>                  &#8627 PiDP</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>M</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>M</font>)
<BR><FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_out_aga</font>) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>N</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#0000cc>in_out_aga</font>) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_out_aga</font>) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#0000cc>type_out_aga</font>) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#0000cc>type_out_aga</font>) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#0000cc>in_out_aga</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><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>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>M</font>))<BR>The graph contains the following edges 1 > 1<P></LI>
<LI><FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_out_aga</font>) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>N</font>)<BR>The graph contains the following edges 1 >= 1<P></LI>
<LI><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>M</font>)<BR>The graph contains the following edges 1 > 1<P></LI>
<LI><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>M</font>)<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><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><pre>                  &#8627 PiDP</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>E</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>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</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_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</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>x1</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_gg</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>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_ggg</font><BR>
<FONT COLOR=#0000cc>U5_ggg</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>U5_ggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_ggg</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_ggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_gga</font><BR>
<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_aga</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_aga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_aga</font><BR>
<FONT COLOR=#0000cc>U5_aga</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>U5_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font><BR>
<FONT COLOR=#0000cc>U6_aga</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_aga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_aga</font><BR>
<FONT COLOR=#0000cc>U2_aga</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>U2_aga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_aga</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>U3_aga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_GGG</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>U5_GGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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 UndefinedPredicateHandlerProof</pre><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><pre>                  &#8627 PiDP</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>E</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>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_gg</font><BR>
<FONT COLOR=#0000cc>U5_GGG</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>U5_GGG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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 UndefinedPredicateHandlerProof</pre><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><pre>                  &#8627 PiDP</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>) &#8594; <FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><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>U5_GGG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>) &#8594; <FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)<BR>The graph contains the following edges 2 >= 1, 3 >= 2, 1 >= 3<P></LI>
<LI><FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))<BR>The graph contains the following edges 1 > 1, 3 >= 1, 1 > 2, 2 >= 3<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><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 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</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_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</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>x1</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_gga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_gg</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>x2</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_ggg</font><BR>
<FONT COLOR=#0000cc>U5_ggg</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>U5_ggg</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_ggg</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_ggg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_gga</font><BR>
<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_aga</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_aga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_aga</font><BR>
<FONT COLOR=#0000cc>U5_aga</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>U5_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font><BR>
<FONT COLOR=#0000cc>U6_aga</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_aga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_aga</font><BR>
<FONT COLOR=#0000cc>U2_aga</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>U2_aga</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_aga</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>U3_aga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TYPE_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 UndefinedPredicateHandlerProof</pre><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 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_out_gga</font>) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#0000cc>in_out_ggg</font>) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_ggg</font>) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#0000cc>in_out_aga</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#0000cc>in_out_aga</font>) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#0000cc>type_out_aga</font>) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_out_aga</font>) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#0000cc>type_out_aga</font>) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#0000cc>type_out_aga</font>) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_out_gga</font>) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#0000cc>type_out_gga</font>) &#8594; <FONT COLOR=#0000cc>type_out_gga</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><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>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>)<BR>The graph contains the following edges 1 >= 1, 2 > 2<P></LI>
<LI><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>))<BR>The graph contains the following edges 1 >= 1, 2 > 2<P></LI>
<LI><FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_out_gga</font>) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>)<BR>The graph contains the following edges 1 >= 1, 2 >= 2<P></LI></UL><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>type_in</font>: (b,b,f) (f,b,f)
<BR><FONT COLOR=#0000cc>in_in</font>: (b,b,f) (b,b,b) (f,b,f)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</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_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</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>x1</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_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>)<BR>
<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_ggg</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>U5_ggg</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>)<BR>
<FONT COLOR=#0000cc>U6_ggg</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_ggg</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>)<BR>
<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_aga</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_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</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>U5_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_aga</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_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_aga</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>U2_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_aga</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>U3_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</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 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</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>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</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_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</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>x1</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_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>)<BR>
<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_ggg</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>U5_ggg</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>)<BR>
<FONT COLOR=#0000cc>U6_ggg</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_ggg</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>)<BR>
<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_aga</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_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</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>U5_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_aga</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_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_aga</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>U2_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_aga</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>U3_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</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>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>\==_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U5_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>\==_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>\==_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</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_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</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>x1</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_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>)<BR>
<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_ggg</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>U5_ggg</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>)<BR>
<FONT COLOR=#0000cc>U6_ggg</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_ggg</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>)<BR>
<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_aga</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_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</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>U5_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_aga</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_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_aga</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>U2_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_aga</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>U3_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_AGA</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_AGA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_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>)<BR>
<FONT COLOR=#0000cc>U1_AGA</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_AGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>\==_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_AGA</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>U5_AGA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_GGG</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_GGG</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>)<BR>
<FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_GGG</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>U5_GGG</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>)<BR>
<FONT COLOR=#0000cc>U2_AGA</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>U2_AGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_GGA</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_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_AGA</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>U3_AGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</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>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>\==_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U5_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_GGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>\==_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>\==_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</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_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</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>x1</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_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>)<BR>
<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_ggg</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>U5_ggg</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>)<BR>
<FONT COLOR=#0000cc>U6_ggg</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_ggg</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>)<BR>
<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_aga</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_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</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>U5_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_aga</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_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_aga</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>U2_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_aga</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>U3_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_AGA</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_AGA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_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>)<BR>
<FONT COLOR=#0000cc>U1_AGA</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_AGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>\==_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_AGA</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>U5_AGA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_GGG</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_GGG</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>)<BR>
<FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_GGG</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>U5_GGG</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>)<BR>
<FONT COLOR=#0000cc>U2_AGA</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>U2_AGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_GGA</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_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>IN_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_AGA</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>U3_AGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 4 SCCs with 17 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</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><pre>                  &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>E</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>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</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_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</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>x1</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_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>)<BR>
<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_ggg</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>U5_ggg</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>)<BR>
<FONT COLOR=#0000cc>U6_ggg</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_ggg</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>)<BR>
<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_aga</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_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</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>U5_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_aga</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_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_aga</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>U2_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_aga</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>U3_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_AGA</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>U5_AGA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</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><pre>                  &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>E</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>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_AGA</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>U5_AGA</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</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 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</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 Rewriting</pre><pre>                  &#8627 PiDP</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>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>)) at position [1] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</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 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</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>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</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 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 UsableRulesProof</pre><pre>                                  &#8627 <B>QDP</B></pre><pre>                                    &#8627 QReductionProof</pre><pre>                  &#8627 PiDP</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>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>R is empty.<BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</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 QDP</pre><pre>                            &#8627 Rewriting</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 UsableRulesProof</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 QReductionProof</pre><pre>                                      &#8627 <B>QDP</B></pre><pre>                                        &#8627 NonTerminationProof</pre><pre>                  &#8627 PiDP</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>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_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>We used the non-termination processor [17] to show that the DP problem is infinite.<BR>Found a loop by narrowing to the left:<BR><BR>The TRS P consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:none<BR><BR><BR>s = <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X'</font>) evaluates to  t =<FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X'</font>)<BR><BR>Thus s starts an infinite chain as s semiunifies with t with the following substitutions:<BR><UL><LI> Semiunifier: [ ]</LI>
<LI> Matcher: [ ]</LI></UL><BR><BR><hr><BR><B>Rewriting sequence</B><BR><BR><B><FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X'</font>)</B> &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X'</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X'</font>))<BR>with rule <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X''</font>) &#8594; <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X''</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X''</font>)) at position [] and matcher [<FONT COLOR=#cc0000>X''</font> / <FONT COLOR=#cc0000>X'</font>]<BR><BR><B><FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X'</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X'</font>))</B> &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X'</font>)<BR>with rule <FONT COLOR=#0000cc>U5_AGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_AGA</font>(<FONT COLOR=#cc0000>X</font>)<BR><BR>Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence<BR><BR><BR>All these steps are and every following step will be a correct step w.r.t to Q.<BR><BR><BR><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</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><pre>                  &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</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_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</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>x1</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_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>)<BR>
<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_ggg</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>U5_ggg</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>)<BR>
<FONT COLOR=#0000cc>U6_ggg</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_ggg</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>)<BR>
<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_aga</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_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</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>U5_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_aga</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_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_aga</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>U2_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_aga</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>U3_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_AGA</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>U2_AGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TYPE_IN_AGA</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 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</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><pre>                  &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>var</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>x1</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_aga</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_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</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>U5_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_aga</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_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_aga</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>U2_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_aga</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>U3_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_AGA</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>U2_AGA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TYPE_IN_AGA</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 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</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><pre>                  &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>M</font>)
<BR><FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><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>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>M</font>))<BR>The graph contains the following edges 1 > 1, 1 > 2<P></LI>
<LI><FONT COLOR=#0000cc>U2_AGA</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>N</font>)<BR>The graph contains the following edges 2 >= 1<P></LI>
<LI><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>M</font>)<BR>The graph contains the following edges 1 > 1<P></LI>
<LI><FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_AGA</font>(<FONT COLOR=#cc0000>M</font>)<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</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><pre>                  &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>E</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>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</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_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</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>x1</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_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>)<BR>
<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_ggg</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>U5_ggg</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>)<BR>
<FONT COLOR=#0000cc>U6_ggg</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_ggg</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>)<BR>
<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_aga</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_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</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>U5_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_aga</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_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_aga</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>U2_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_aga</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>U3_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U5_GGG</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>U5_GGG</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>)<BR>
<FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</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><pre>                  &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>E</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>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)</BLOCKQUOTE><BR>Pi is empty.<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 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</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><pre>                  &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>E</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>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><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>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)<BR>The graph contains the following edges 3 >= 1, 4 >= 2, 5 > 2, 2 >= 3<P></LI>
<LI><FONT COLOR=#0000cc>IN_IN_GGG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_GGG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))<BR>The graph contains the following edges 1 > 1, 1 > 2, 3 >= 2, 1 > 3, 2 >= 4<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</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 <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>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>S</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>), <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>arrow</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#cc0000>T</font>))) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>), <FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gga</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_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x1</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>x1</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_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>U5_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>)<BR>
<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_ggg</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>U5_ggg</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>)<BR>
<FONT COLOR=#0000cc>U6_ggg</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_ggg</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>)<BR>
<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_aga</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_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_aga</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>U5_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_aga</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_aga</font>(<FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_aga</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>U2_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_aga</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>U3_aga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_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>U3_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_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>U2_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>TYPE_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 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</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 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>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>)
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>) &#8594; <FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_ggg</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>E</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>\==_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>in_out_aga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>var</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>type_out_aga</font>(<FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>lambda</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>))
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>in_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>\==_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>in_in_ggg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U5_ggg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U6_ggg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)
<BR><FONT COLOR=#0000cc>U1_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>type_in_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>in_in_aga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>\==_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U6_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U2_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U3_aga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>U2_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)
<BR><FONT COLOR=#0000cc>U3_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)</BLOCKQUOTE><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>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>)<BR>The graph contains the following edges 1 >= 1, 2 > 2<P></LI>
<LI><FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#0000cc>apply</font>(<FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>)) &#8594; <FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_in_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>))<BR>The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3<P></LI>
<LI><FONT COLOR=#0000cc>U2_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#cc0000>N</font>, <FONT COLOR=#0000cc>type_out_gga</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>TYPE_IN_GGA</font>(<FONT COLOR=#cc0000>E</font>, <FONT COLOR=#cc0000>N</font>)<BR>The graph contains the following edges 1 >= 1, 4 > 1, 3 >= 2<P></LI></UL><BR><BR></body>

