MAYBE
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<title>Left Termination proof of ../tpdb/LP/BCGGV05/slowsort-fb.pl</title>
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<BR><B>Left Termination</B> of the query pattern
ss_in_2(a, g)
w.r.t. the given <I>Prolog program</I> could not be shown:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>ss(Xs, Ys)&#160;:-&#160;','(perm(Xs, Ys), ordered(Ys)).<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>ordered([]).<BR>ordered(.(X, [])).<BR>ordered(.(X, .(Y, Xs)))&#160;:-&#160;','(less(X, s(Y)), ordered(.(Y, Xs))).<BR>less(0, s(X)).<BR>less(s(X), s(Y))&#160;:-&#160;less(X, Y).<BR><BR>Queries:<BR><BR>ss(a,g).<BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>ss_in</font>: (f,b)
<BR><FONT COLOR=#0000cc>perm_in</font>: (f,b)
<BR><FONT COLOR=#0000cc>app_in</font>: (f,f,f)
<BR><FONT COLOR=#0000cc>ordered_in</font>: (b)
<BR><FONT COLOR=#0000cc>less_in</font>: (b,b)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aaa</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>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aaa</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>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ag</font><BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aaa</font><BR>
<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_out_g</font><BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_out_ag</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><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aaa</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>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aaa</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>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ag</font><BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aaa</font><BR>
<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_out_g</font><BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_out_ag</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>SS_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>SS_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_AG</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_AAA</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_AAA</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>U6_AAA</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>APP_IN_AAA</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_AAA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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_AAA</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aaa</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>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aaa</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>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ag</font><BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aaa</font><BR>
<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_out_g</font><BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_out_ag</font><BR>
<FONT COLOR=#0000cc>APP_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_IN_AAA</font><BR>
<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_AAA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_G</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>SS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>SS_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<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><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>SS_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>SS_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_AG</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_AAA</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_AAA</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>U6_AAA</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>APP_IN_AAA</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_AAA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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_AAA</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aaa</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>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aaa</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>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ag</font><BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aaa</font><BR>
<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_out_g</font><BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_out_ag</font><BR>
<FONT COLOR=#0000cc>APP_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_IN_AAA</font><BR>
<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_AAA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_G</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>SS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>SS_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 4 SCCs with 11 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aaa</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>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aaa</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>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ag</font><BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aaa</font><BR>
<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_out_g</font><BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_out_ag</font><BR>
<FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>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><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>Pi is empty.<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 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><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 > 1, 2 > 2<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aaa</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>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aaa</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>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ag</font><BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aaa</font><BR>
<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_out_g</font><BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_out_ag</font><BR>
<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ORDERED_IN_G</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 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ORDERED_IN_G</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 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 UsableRulesReductionPairsProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>) &#8594; <FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#0000cc>less_out_gg</font>) &#8594; <FONT COLOR=#0000cc>less_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.<BR><BR>The following dependency pairs can be deleted:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>) &#8594; <FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE>The following rules are removed from R:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font></BLOCKQUOTE>Used ordering: POLO with Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>.</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 2&middot;x<SUB>1</SUB> + 2&middot;x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>0</font></B>) = 2<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>ORDERED_IN_G</font>(x<SUB>1</SUB>)</B>) = 2 + x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U7_G</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 2 + 2&middot;x<SUB>1</SUB> + 2&middot;x<SUB>2</SUB> + x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U9_gg</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>less_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>1</SUB> + 2&middot;x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>less_out_gg</font></B>) = 2<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>s</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 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 UsableRulesReductionPairsProof</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 DependencyGraphProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#0000cc>less_out_gg</font>) &#8594; <FONT COLOR=#0000cc>less_out_gg</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APP_IN_AAA</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_AAA</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>ss_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aaa</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>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aaa</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>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ag</font><BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aaa</font><BR>
<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_out_g</font><BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_out_ag</font><BR>
<FONT COLOR=#0000cc>APP_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_IN_AAA</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><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APP_IN_AAA</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_AAA</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>APP_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_IN_AAA</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 NonTerminationProof</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APP_IN_AAA</font> &#8594; <FONT COLOR=#0000cc>APP_IN_AAA</font></BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>We used the non-termination processor [17] to show that the DP problem is infinite.<BR>Found a loop by semiunifying a rule from P directly.<BR><BR>The TRS P consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>APP_IN_AAA</font> &#8594; <FONT COLOR=#0000cc>APP_IN_AAA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:none<BR><BR><BR>s = <FONT COLOR=#0000cc>APP_IN_AAA</font> evaluates to  t =<FONT COLOR=#0000cc>APP_IN_AAA</font><BR><BR>Thus s starts an infinite chain as s semiunifies with t with the following substitutions:<BR><UL><LI> Semiunifier: [ ]</LI>
<LI> Matcher: [ ]</LI></UL><BR><BR><hr><BR><B>Rewriting sequence</B><BR><BR>The DP semiunifies directly so there is only one rewrite step from APP_IN_AAA to APP_IN_AAA.<BR><BR><BR><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aaa</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>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aaa</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>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ag</font><BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aaa</font><BR>
<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_out_g</font><BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_gg</font><BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_out_ag</font><BR>
<FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aaa</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>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aaa</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>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aaa</font><BR>
<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 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 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>)
<BR><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>)
<BR><FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>) &#8594; <FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>app_in_aaa</font> &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>
<BR><FONT COLOR=#0000cc>app_in_aaa</font> &#8594; <FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#0000cc>app_in_aaa</font>)
<BR><FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#0000cc>app_out_aaa</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>app_in_aaa</font>
<BR><FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>)<BR>The graph contains the following edges 1 > 1<P></LI>
<LI><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>)<BR>The graph contains the following edges 1 >= 1<P></LI>
<LI><FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>) &#8594; <FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Ys</font>)<BR>The graph contains the following edges 1 >= 1<P></LI></UL><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>ss_in</font>: (f,b)
<BR><FONT COLOR=#0000cc>perm_in</font>: (f,b)
<BR><FONT COLOR=#0000cc>app_in</font>: (f,f,f)
<BR><FONT COLOR=#0000cc>ordered_in</font>: (b)
<BR><FONT COLOR=#0000cc>less_in</font>: (b,b)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aaa</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>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aaa</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>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aaa</font><BR>
<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 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>ss_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aaa</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>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aaa</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>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aaa</font><BR>
<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR><BR>Using Dependency Pairs [1,30] we result in the following initial DP problem:<BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>SS_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>SS_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_AG</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_AAA</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_AAA</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>U6_AAA</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>APP_IN_AAA</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_AAA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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_AAA</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aaa</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>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aaa</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>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aaa</font><BR>
<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>APP_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_IN_AAA</font><BR>
<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_AAA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>SS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>SS_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 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>SS_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>SS_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_AG</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_AAA</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_AAA</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>U6_AAA</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>APP_IN_AAA</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_AAA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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_AAA</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aaa</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>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aaa</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>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aaa</font><BR>
<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>APP_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_IN_AAA</font><BR>
<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_AAA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U8_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>SS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>SS_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 4 SCCs with 11 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aaa</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>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aaa</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>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aaa</font><BR>
<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>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 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>Pi is empty.<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 > 1, 2 > 2<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aaa</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>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aaa</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>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aaa</font><BR>
<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ORDERED_IN_G</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 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))</BLOCKQUOTE><BR>Pi is empty.<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 RuleRemovalProof</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>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</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>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))</BLOCKQUOTE><BR>Strictly oriented rules of the TRS R:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>Used ordering: POLO with Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>.</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 2&middot;x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>0</font></B>) = 2<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>ORDERED_IN_G</font>(x<SUB>1</SUB>)</B>) = 2&middot;x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U7_G</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>, x<SUB>4</SUB>)</B>) = x<SUB>1</SUB> + 2&middot;x<SUB>2</SUB> + 2&middot;x<SUB>3</SUB> + x<SUB>4</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U9_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = x<SUB>1</SUB> + x<SUB>2</SUB> + x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>less_in_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 2&middot;x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>less_out_gg</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>s</font>(x<SUB>1</SUB>)</B>) = 2&middot;x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 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 QDP</pre><pre>                        &#8627 RuleRemovalProof</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 DependencyGraphProof</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>ORDERED_IN_G</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_G</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</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 1 less node.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APP_IN_AAA</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_AAA</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>ss_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aaa</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>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aaa</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>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aaa</font><BR>
<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>APP_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_IN_AAA</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 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APP_IN_AAA</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_AAA</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>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>APP_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_IN_AAA</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 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 NonTerminationProof</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_AAA</font> &#8594; <FONT COLOR=#0000cc>APP_IN_AAA</font></BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>We used the non-termination processor [17] to show that the DP problem is infinite.<BR>Found a loop by semiunifying a rule from P directly.<BR><BR>The TRS P consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>APP_IN_AAA</font> &#8594; <FONT COLOR=#0000cc>APP_IN_AAA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:none<BR><BR><BR>s = <FONT COLOR=#0000cc>APP_IN_AAA</font> evaluates to  t =<FONT COLOR=#0000cc>APP_IN_AAA</font><BR><BR>Thus s starts an infinite chain as s semiunifies with t with the following substitutions:<BR><UL><LI> Semiunifier: [ ]</LI>
<LI> Matcher: [ ]</LI></UL><BR><BR><hr><BR><B>Rewriting sequence</B><BR><BR>The DP semiunifies directly so there is only one rewrite step from APP_IN_AAA to APP_IN_AAA.<BR><BR><BR><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aaa</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>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aaa</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>U3_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>Ys</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))) &#8594; <FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>perm_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aaa</font><BR>
<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>ordered_out_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U7_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U9_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>ss_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</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 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 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>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Zs</font>, <FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</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>U3_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</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>U4_AG</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>Zs</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_in_aaa</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>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U6_aaa</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_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_out_aaa</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>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_out_aaa</font><BR>
<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>) &#8594; <FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Ys</font>)
<BR><FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>)
<BR><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>app_in_aaa</font> &#8594; <FONT COLOR=#0000cc>app_out_aaa</font>
<BR><FONT COLOR=#0000cc>app_in_aaa</font> &#8594; <FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#0000cc>app_in_aaa</font>)
<BR><FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#0000cc>app_out_aaa</font>) &#8594; <FONT COLOR=#0000cc>app_out_aaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>app_in_aaa</font>
<BR><FONT COLOR=#0000cc>U6_aaa</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>)) &#8594; <FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>)<BR>The graph contains the following edges 1 > 1, 1 > 2<P></LI>
<LI><FONT COLOR=#0000cc>U3_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_in_aaa</font>)<BR>The graph contains the following edges 1 >= 1, 2 >= 2<P></LI>
<LI><FONT COLOR=#0000cc>U4_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#0000cc>app_out_aaa</font>) &#8594; <FONT COLOR=#0000cc>PERM_IN_AG</font>(<FONT COLOR=#cc0000>Ys</font>)<BR>The graph contains the following edges 2 >= 1<P></LI></UL><BR><BR></body>

