YES
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<title>Left Termination proof of ../tpdb/LP/talp/talp/vangelder.pl</title>
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<BR><B>Left Termination</B> of the query pattern
q_in_2(g, 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>e(a, b).<BR>q(X, Y)&#160;:-&#160;e(X, Y).<BR>q(X, f(f(X)))&#160;:-&#160;','(p(X, f(f(X))), q(X, f(X))).<BR>q(X, f(f(Y)))&#160;:-&#160;p(X, f(Y)).<BR>p(X, Y)&#160;:-&#160;e(X, Y).<BR>p(X, f(Y))&#160;:-&#160;','(r(X, f(Y)), p(X, Y)).<BR>r(X, Y)&#160;:-&#160;e(X, Y).<BR>r(X, f(Y))&#160;:-&#160;','(q(X, Y), r(X, Y)).<BR>r(f(X), f(X))&#160;:-&#160;t(f(X), f(X)).<BR>t(X, Y)&#160;:-&#160;e(X, Y).<BR>t(f(X), f(Y))&#160;:-&#160;','(q(f(X), f(Y)), t(X, Y)).<BR><BR>Queries:<BR><BR>q(g,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>q_in</font>: (b,b)
<BR><FONT COLOR=#0000cc>p_in</font>: (b,b)
<BR><FONT COLOR=#0000cc>r_in</font>: (b,b)
<BR><FONT COLOR=#0000cc>t_in</font>: (b,b)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>b</font>) &#8594; <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>b</font>)
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))) &#8594; <FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>a</font>&#160; = &#160;<FONT COLOR=#0000cc>a</font><BR>
<FONT COLOR=#0000cc>b</font>&#160; = &#160;<FONT COLOR=#0000cc>b</font><BR>
<FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_out_gg</font><BR>
<FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>q_out_gg</font><BR>
<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_out_gg</font><BR>
<FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>r_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_out_gg</font><BR>
<FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x2</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>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>b</font>) &#8594; <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>b</font>)
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))) &#8594; <FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>a</font>&#160; = &#160;<FONT COLOR=#0000cc>a</font><BR>
<FONT COLOR=#0000cc>b</font>&#160; = &#160;<FONT COLOR=#0000cc>b</font><BR>
<FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_out_gg</font><BR>
<FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>q_out_gg</font><BR>
<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_out_gg</font><BR>
<FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>r_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_out_gg</font><BR>
<FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x2</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>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>E_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>E_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U8_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>E_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U4_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U10_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U11_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U12_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>E_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U13_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U13_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U14_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U13_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U7_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))) &#8594; <FONT COLOR=#0000cc>U3_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>b</font>) &#8594; <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>b</font>)
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))) &#8594; <FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>a</font>&#160; = &#160;<FONT COLOR=#0000cc>a</font><BR>
<FONT COLOR=#0000cc>b</font>&#160; = &#160;<FONT COLOR=#0000cc>b</font><BR>
<FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_out_gg</font><BR>
<FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>q_out_gg</font><BR>
<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_out_gg</font><BR>
<FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>r_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_out_gg</font><BR>
<FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U11_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U11_GG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U12_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U12_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>E_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>E_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U13_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U13_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U10_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U10_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U14_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U14_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U7_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GG</font>(<FONT COLOR=#cc0000>x3</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>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>E_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>E_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U8_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>E_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U4_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U10_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U11_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U12_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>E_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U13_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U13_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U14_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U13_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U7_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))) &#8594; <FONT COLOR=#0000cc>U3_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>b</font>) &#8594; <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>b</font>)
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))) &#8594; <FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>a</font>&#160; = &#160;<FONT COLOR=#0000cc>a</font><BR>
<FONT COLOR=#0000cc>b</font>&#160; = &#160;<FONT COLOR=#0000cc>b</font><BR>
<FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_out_gg</font><BR>
<FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>q_out_gg</font><BR>
<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_out_gg</font><BR>
<FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>r_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_out_gg</font><BR>
<FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U11_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U11_GG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U12_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U12_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>E_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>E_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U13_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U13_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U10_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U10_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U14_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U14_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U7_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 1 SCC with 14 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 <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>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U13_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U13_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>b</font>) &#8594; <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>b</font>)
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))) &#8594; <FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>a</font>&#160; = &#160;<FONT COLOR=#0000cc>a</font><BR>
<FONT COLOR=#0000cc>b</font>&#160; = &#160;<FONT COLOR=#0000cc>b</font><BR>
<FONT COLOR=#0000cc>e_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>e_out_gg</font><BR>
<FONT COLOR=#0000cc>q_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>q_out_gg</font><BR>
<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>p_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>p_out_gg</font><BR>
<FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>r_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>r_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>t_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>t_out_gg</font><BR>
<FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U13_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U13_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>T_IN_GG</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 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>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U13_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>U13_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>b</font>) &#8594; <FONT COLOR=#0000cc>e_out_gg</font>
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#0000cc>t_out_gg</font>) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>
<BR><FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#0000cc>t_out_gg</font>) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#0000cc>r_out_gg</font>) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>q_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_gg</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>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U13_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>U13_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE>Used ordering:  Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>P_IN_GG</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>Q_IN_GG</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>R_IN_GG</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>T_IN_GG</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>U10_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U11_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U12_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U13_GG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U13_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U14_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_GG</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_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U5_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U6_GG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U6_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U7_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U8_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U9_GG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U9_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>a</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>b</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>e_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>e_out_gg</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>f</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>p_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>p_out_gg</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>q_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>q_out_gg</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>r_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>r_out_gg</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>t_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>t_out_gg</font></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 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>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>U13_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>b</font>) &#8594; <FONT COLOR=#0000cc>e_out_gg</font>
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#0000cc>t_out_gg</font>) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>
<BR><FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#0000cc>t_out_gg</font>) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#0000cc>r_out_gg</font>) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>q_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_gg</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 1 less node.<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 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 Instantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>b</font>) &#8594; <FONT COLOR=#0000cc>e_out_gg</font>
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#0000cc>t_out_gg</font>) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>
<BR><FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#0000cc>t_out_gg</font>) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#0000cc>r_out_gg</font>) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>q_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>z0</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 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 Instantiation</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>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>z0</font>))
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>b</font>) &#8594; <FONT COLOR=#0000cc>e_out_gg</font>
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#0000cc>t_out_gg</font>) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>
<BR><FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#0000cc>t_out_gg</font>) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#0000cc>r_out_gg</font>) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>q_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_gg</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>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>z0</font>))
<BR><FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE>Used ordering:  Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>P_IN_GG</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>Q_IN_GG</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>R_IN_GG</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>T_IN_GG</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>U10_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U11_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U12_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U13_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U14_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_GG</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_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U3_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U5_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U6_GG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U6_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U7_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U8_gg</font>(x<SUB>1</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U9_GG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U9_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>a</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>b</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>e_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>e_out_gg</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>f</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>p_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>p_out_gg</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>q_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>q_out_gg</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>r_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>r_out_gg</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>t_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>t_out_gg</font></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 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 Instantiation</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>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>U6_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>) &#8594; <FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>z0</font>)) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>z0</font>))
<BR><FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>T_IN_GG</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>P_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>R_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>Q_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>b</font>) &#8594; <FONT COLOR=#0000cc>e_out_gg</font>
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>))))
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>q_out_gg</font>
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#0000cc>e_out_gg</font>) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#0000cc>t_out_gg</font>) &#8594; <FONT COLOR=#0000cc>t_out_gg</font>
<BR><FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#0000cc>t_out_gg</font>) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#0000cc>r_out_gg</font>) &#8594; <FONT COLOR=#0000cc>r_out_gg</font>
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>r_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>p_out_gg</font>
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>p_out_gg</font>) &#8594; <FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>f</font>(<FONT COLOR=#cc0000>X</font>)))
<BR><FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#0000cc>q_out_gg</font>) &#8594; <FONT COLOR=#0000cc>q_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>q_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>e_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>p_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>r_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U8_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>t_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U12_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U13_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U14_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U11_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U10_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>U7_gg</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 7 less nodes.<BR><BR></body>

