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<title>Left Termination proof of ../tpdb/LP/BCGGV05/p_nonlin.pl</title>
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<BR><B>Left Termination</B> of the query pattern
p_in_1(g)
w.r.t. the given <I>Prolog program</I> could successfully be <font color=#00ff00>proven</font>:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>p(.(X, [])).<BR>p(.(s(s(X)), .(Y, Xs)))&#160;:-&#160;','(p(.(X, .(Y, Xs))), ','(mult(X, Y, Z), p(.(Z, Xs)))).<BR>p(.(0, Xs))&#160;:-&#160;p(Xs).<BR>sum(X, 0, X).<BR>sum(X, s(Y), s(Z))&#160;:-&#160;sum(X, Y, Z).<BR>mult(X, 0, 0).<BR>mult(X, s(Y), Z)&#160;:-&#160;','(mult(X, Y, W), sum(W, X, Z)).<BR><BR>Queries:<BR><BR>p(g).<BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>p_in</font>: (b)
<BR><FONT COLOR=#0000cc>mult_in</font>: (b,b,f)
<BR><FONT COLOR=#0000cc>sum_in</font>: (b,b,f)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))) &#8594; <FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_in_g</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>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_out_g</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_g</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_g</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>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_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>U7_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sum_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>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 <B>PiTRS</B></pre><pre>      &#8627 DependencyPairsProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))) &#8594; <FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_in_g</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>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_out_g</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_g</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_g</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>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_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>U7_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sum_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>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x4</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>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U4_G</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))) &#8594; <FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))) &#8594; <FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>)
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U3_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))) &#8594; <FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_in_g</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>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_out_g</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_g</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_g</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>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_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>U7_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sum_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>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>P_IN_G</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>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_G</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_G</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_G</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>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</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>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_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>U7_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_G</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 <B>PiDP</B></pre><pre>          &#8627 DependencyGraphProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U4_G</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))) &#8594; <FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))) &#8594; <FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>)
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>U7_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U5_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U3_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))) &#8594; <FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_in_g</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>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_out_g</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_g</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_g</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>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_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>U7_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sum_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>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>P_IN_G</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>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_G</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_G</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_G</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>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</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>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_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>U7_GGA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_G</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 3 SCCs with 7 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))) &#8594; <FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_in_g</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>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_out_g</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_g</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_g</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>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_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>U7_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sum_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>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>SUM_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 >= 1, 2 > 2<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 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>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))) &#8594; <FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_in_g</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>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_out_g</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_g</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_g</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>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_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>U7_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sum_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>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>MULT_IN_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 >= 1, 2 > 2<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 PiDPToQDPProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))) &#8594; <FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))) &#8594; <FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>), <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_in_g</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>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>p_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_out_g</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_g</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_g</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>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U7_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>U7_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>sum_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>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_G</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_G</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>U2_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 PiDPToQDPProof</pre><pre>                  &#8627 <B>QDP</B></pre><pre>                    &#8627 QDPOrderProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>) &#8594; <FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>p_out_g</font>
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#0000cc>p_out_g</font>) &#8594; <FONT COLOR=#0000cc>p_out_g</font>
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>) &#8594; <FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#0000cc>p_out_g</font>) &#8594; <FONT COLOR=#0000cc>p_out_g</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_g</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>mult_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We use the reduction pair processor [15].<P><BR>The following pairs can be oriented strictly and are deleted.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>) &#8594; <FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))</BLOCKQUOTE>Used ordering:  Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>.</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>0</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>P_IN_G</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_G</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 1 + x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_g</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_G</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_g</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_g</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_g</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U5_gga</font>(x<SUB>1</SUB>)</B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U6_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U7_gga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>[]</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>mult_in_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>mult_out_gga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>p_in_g</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>p_out_g</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>s</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>sum_in_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>sum_out_gga</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR>The following usable rules [17] were oriented:
none<BR><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 PiDPToQDPProof</pre><pre>                  &#8627 QDP</pre><pre>                    &#8627 QDPOrderProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 DependencyGraphProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>) &#8594; <FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>p_out_g</font>
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#0000cc>p_out_g</font>) &#8594; <FONT COLOR=#0000cc>p_out_g</font>
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>) &#8594; <FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#0000cc>p_out_g</font>) &#8594; <FONT COLOR=#0000cc>p_out_g</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_g</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>mult_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 PiDPToQDPProof</pre><pre>                  &#8627 QDP</pre><pre>                    &#8627 QDPOrderProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 DependencyGraphProof</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 UsableRulesProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>p_out_g</font>
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))))
<BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#0000cc>p_out_g</font>) &#8594; <FONT COLOR=#0000cc>p_out_g</font>
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>p_out_g</font>) &#8594; <FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mult_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#0000cc>sum_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>mult_out_gga</font>(<FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#0000cc>p_out_g</font>) &#8594; <FONT COLOR=#0000cc>p_out_g</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_g</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>mult_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_g</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 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 PiDPToQDPProof</pre><pre>                  &#8627 QDP</pre><pre>                    &#8627 QDPOrderProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 DependencyGraphProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 UsableRulesProof</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 QReductionProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))</BLOCKQUOTE><BR>R is empty.<BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>p_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_g</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>mult_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_g</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>p_in_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U1_g</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>mult_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U6_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>sum_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U7_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 PiDPToQDPProof</pre><pre>                  &#8627 QDP</pre><pre>                    &#8627 QDPOrderProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 DependencyGraphProof</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 UsableRulesReductionPairsProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.<BR><BR>The following dependency pairs can be deleted:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))</BLOCKQUOTE>No rules are removed from R.<BR><BR>Used ordering: POLO with Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>.</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 2&middot;x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>P_IN_G</font>(x<SUB>1</SUB>)</B>) = 2&middot;x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>s</font>(x<SUB>1</SUB>)</B>) = 2&middot;x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 PiDPToQDPProof</pre><pre>                  &#8627 QDP</pre><pre>                    &#8627 QDPOrderProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 DependencyGraphProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 UsableRulesProof</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 QReductionProof</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 UsableRulesReductionPairsProof</pre><pre>                                      &#8627 <B>QDP</B></pre><pre>                                        &#8627 PisEmptyProof</pre><BR>Q DP problem:<BR>P is empty.<BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>The TRS P is empty. Hence, there is no (P,Q,R) chain.<BR><BR></body>

