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<title>Left Termination proof of ../tpdb/LP/BCGGV05/permutation-bf.pl</title>
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<BR><B>Left Termination</B> of the query pattern
perm_in_2(g, a)
w.r.t. the given <I>Prolog program</I> could successfully be <font color=#00ff00>proven</font>:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>perm([], []).<BR>perm(Xs, .(X, Ys))&#160;:-&#160;','(app(X1s, .(X, X2s), Xs), ','(app(X1s, X2s, Zs), perm(Zs, Ys))).<BR>app([], X, X).<BR>app(.(X, Xs), Ys, .(X, Zs))&#160;:-&#160;app(Xs, Ys, Zs).<BR><BR>Queries:<BR><BR>perm(g,a).<BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>perm_in</font>: (b,f)
<BR><FONT COLOR=#0000cc>app_in</font>: (f,f,b) (b,b,f)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<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>perm_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<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>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_AAG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_AAG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AAG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GGA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 <B>PiDP</B></pre><pre>          &#8627 DependencyGraphProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>)
<BR><FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_AAG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_GGA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_AAG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AAG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GGA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 3 SCCs with 5 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>APP_IN_GGA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)<BR>The graph contains the following edges 1 > 1, 2 >= 2<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#cc0000>Zs</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>APP_IN_AAG</font>(<FONT COLOR=#cc0000>Zs</font>)<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ga</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>), <FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Zs</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 RuleRemovalProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X2s</font>))) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
<BR>Strictly oriented dependency pairs:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X2s</font>))) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))</BLOCKQUOTE><BR><BR>Used ordering: POLO with Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>.</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>PERM_IN_GA</font>(x<SUB>1</SUB>)</B>) = 2&middot;x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_GA</font>(x<SUB>1</SUB>)</B>) = 2&middot;x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_GA</font>(x<SUB>1</SUB>)</B>) = 2&middot;x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_aag</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_gga</font>(x<SUB>1</SUB>)</B>) = 1 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>[]</font></B>) = 0<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>app_in_aag</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>app_in_gga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>app_out_aag</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>app_out_gga</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 RuleRemovalProof</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>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Xs</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>Xs</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_GA</font>(<FONT COLOR=#cc0000>Zs</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_gga</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_aag</font>(<FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>app_in_gga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>app_in_aag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_gga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_aag</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 2 less nodes.<BR><BR></body>

