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<title>Left Termination proof of ../tpdb/LP/BCGGV05/g.pl</title>
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<BR><B>Left Termination</B> of the query pattern
g_in_1(a)
w.r.t. the given <I>Prolog program</I> could not be shown:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>g(W)&#160;:-&#160;','(eq(X, .(.(a, []), .(.(R, []), []))), ','(eq(Y, .(.(S, .(c, [])), .([], []))), ','(app_1(X, Y, Z), ','(eq(Z, .(U, V)), app_2(U, U, W))))).<BR>app_1([], X, X).<BR>app_1(.(X, Xs), Ys, .(X, Zs))&#160;:-&#160;app_1(Xs, Ys, Zs).<BR>app_2([], X, X).<BR>app_2(.(X, Xs), Ys, .(X, Zs))&#160;:-&#160;app_2(Xs, Ys, Zs).<BR>eq(X, X).<BR><BR>Queries:<BR><BR>g(a).<BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>g_in</font>: (f)
<BR><FONT COLOR=#0000cc>app_1_in</font>: (f,f,f)
<BR><FONT COLOR=#0000cc>app_2_in</font>: (f,f,f)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>g_in_a</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_1_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_1_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_1_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>)))
<BR><FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_2_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>U7_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_2_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U7_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_2_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_2_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>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>W</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>g_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_in_a</font><BR>
<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_aa</font><BR>
<FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_aa</font><BR>
<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_out_aaa</font><BR>
<FONT COLOR=#0000cc>U7_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>U7_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_out_a</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>g_in_a</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_1_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_1_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_1_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>)))
<BR><FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_2_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>U7_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_2_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U7_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_2_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_2_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>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>W</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>g_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_in_a</font><BR>
<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_aa</font><BR>
<FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_aa</font><BR>
<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_out_aaa</font><BR>
<FONT COLOR=#0000cc>U7_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>U7_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_out_a</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>G_IN_A</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>G_IN_A</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>EQ_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U2_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>EQ_IN_AA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U2_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U2_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>APP_1_IN_AAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>APP_1_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_1_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>APP_1_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_1_IN_AAA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>)))
<BR><FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>EQ_IN_AA</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))
<BR><FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>U5_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>APP_2_IN_AAA</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>)
<BR><FONT COLOR=#0000cc>APP_2_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>U7_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_2_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>APP_2_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_2_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>g_in_a</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_1_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_1_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_1_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>)))
<BR><FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_2_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>U7_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_2_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U7_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_2_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_2_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>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>W</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>g_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_in_a</font><BR>
<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_aa</font><BR>
<FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_aa</font><BR>
<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_out_aaa</font><BR>
<FONT COLOR=#0000cc>U7_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>U7_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_out_a</font><BR>
<FONT COLOR=#0000cc>APP_1_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_1_IN_AAA</font><BR>
<FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>G_IN_A</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>G_IN_A</font><BR>
<FONT COLOR=#0000cc>U7_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>U7_AAA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>EQ_IN_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>EQ_IN_AA</font><BR>
<FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_A</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>APP_2_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_2_IN_AAA</font><BR>
<FONT COLOR=#0000cc>U5_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_A</font>(<FONT COLOR=#cc0000>x2</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>G_IN_A</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>G_IN_A</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>EQ_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U2_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>EQ_IN_AA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U2_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U2_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>APP_1_IN_AAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>APP_1_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_1_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>APP_1_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_1_IN_AAA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>)))
<BR><FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>EQ_IN_AA</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))
<BR><FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>U5_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>APP_2_IN_AAA</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>)
<BR><FONT COLOR=#0000cc>APP_2_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>U7_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_2_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>APP_2_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_2_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>g_in_a</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_1_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_1_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_1_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>)))
<BR><FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_2_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>U7_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_2_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U7_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_2_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_2_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>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>W</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>g_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_in_a</font><BR>
<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_aa</font><BR>
<FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_aa</font><BR>
<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_out_aaa</font><BR>
<FONT COLOR=#0000cc>U7_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>U7_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_out_a</font><BR>
<FONT COLOR=#0000cc>APP_1_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_1_IN_AAA</font><BR>
<FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>G_IN_A</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>G_IN_A</font><BR>
<FONT COLOR=#0000cc>U7_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>U7_AAA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>EQ_IN_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>EQ_IN_AA</font><BR>
<FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_A</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>APP_2_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_2_IN_AAA</font><BR>
<FONT COLOR=#0000cc>U5_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_A</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 2 SCCs with 12 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 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APP_2_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_2_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>g_in_a</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_1_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_1_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_1_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>)))
<BR><FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_2_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>U7_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_2_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U7_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_2_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_2_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>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>W</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>g_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_in_a</font><BR>
<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_aa</font><BR>
<FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_aa</font><BR>
<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_out_aaa</font><BR>
<FONT COLOR=#0000cc>U7_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>U7_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_out_a</font><BR>
<FONT COLOR=#0000cc>APP_2_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_2_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 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_2_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_2_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>APP_2_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_2_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 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_2_IN_AAA</font> &#8594; <FONT COLOR=#0000cc>APP_2_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_2_IN_AAA</font> &#8594; <FONT COLOR=#0000cc>APP_2_IN_AAA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:none<BR><BR><BR>s = <FONT COLOR=#0000cc>APP_2_IN_AAA</font> evaluates to  t =<FONT COLOR=#0000cc>APP_2_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_2_IN_AAA to APP_2_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 <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>APP_1_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_1_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>g_in_a</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_1_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_1_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_1_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>)))
<BR><FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_2_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>U7_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_2_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U7_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_2_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_2_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>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>W</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>g_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_in_a</font><BR>
<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_aa</font><BR>
<FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_aa</font><BR>
<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_out_aaa</font><BR>
<FONT COLOR=#0000cc>U7_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>U7_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_out_a</font><BR>
<FONT COLOR=#0000cc>APP_1_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_1_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 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>APP_1_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_1_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>APP_1_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_1_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 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 NonTerminationProof</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_1_IN_AAA</font> &#8594; <FONT COLOR=#0000cc>APP_1_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_1_IN_AAA</font> &#8594; <FONT COLOR=#0000cc>APP_1_IN_AAA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:none<BR><BR><BR>s = <FONT COLOR=#0000cc>APP_1_IN_AAA</font> evaluates to  t =<FONT COLOR=#0000cc>APP_1_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_1_IN_AAA to APP_1_IN_AAA.<BR><BR><BR><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>g_in</font>: (f)
<BR><FONT COLOR=#0000cc>app_1_in</font>: (f,f,f)
<BR><FONT COLOR=#0000cc>app_2_in</font>: (f,f,f)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>g_in_a</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_1_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_1_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_1_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>)))
<BR><FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_2_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>U7_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_2_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U7_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_2_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_2_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>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>W</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>g_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_in_a</font><BR>
<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_aa</font><BR>
<FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_aa</font><BR>
<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_out_aaa</font><BR>
<FONT COLOR=#0000cc>U7_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>U7_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_out_a</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>g_in_a</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_1_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_1_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_1_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>)))
<BR><FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_2_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>U7_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_2_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U7_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_2_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_2_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>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>W</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>g_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_in_a</font><BR>
<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_aa</font><BR>
<FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_aa</font><BR>
<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_out_aaa</font><BR>
<FONT COLOR=#0000cc>U7_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>U7_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_out_a</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>G_IN_A</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>G_IN_A</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>EQ_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U2_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>EQ_IN_AA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U2_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U2_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>APP_1_IN_AAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>APP_1_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_1_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>APP_1_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_1_IN_AAA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>)))
<BR><FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>EQ_IN_AA</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))
<BR><FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>U5_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>APP_2_IN_AAA</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>)
<BR><FONT COLOR=#0000cc>APP_2_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>U7_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_2_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>APP_2_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_2_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>g_in_a</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_1_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_1_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_1_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>)))
<BR><FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_2_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>U7_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_2_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U7_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_2_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_2_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>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>W</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>g_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_in_a</font><BR>
<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_aa</font><BR>
<FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_aa</font><BR>
<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_out_aaa</font><BR>
<FONT COLOR=#0000cc>U7_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>U7_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_out_a</font><BR>
<FONT COLOR=#0000cc>APP_1_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_1_IN_AAA</font><BR>
<FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>G_IN_A</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>G_IN_A</font><BR>
<FONT COLOR=#0000cc>U7_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>U7_AAA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>EQ_IN_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>EQ_IN_AA</font><BR>
<FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_A</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>APP_2_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_2_IN_AAA</font><BR>
<FONT COLOR=#0000cc>U5_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_A</font>(<FONT COLOR=#cc0000>x2</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>G_IN_A</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>G_IN_A</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>EQ_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U2_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>EQ_IN_AA</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>U2_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>U2_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>APP_1_IN_AAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)
<BR><FONT COLOR=#0000cc>APP_1_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_1_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>APP_1_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_1_IN_AAA</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)
<BR><FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>)))
<BR><FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>EQ_IN_AA</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))
<BR><FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>U5_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>APP_2_IN_AAA</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>)
<BR><FONT COLOR=#0000cc>APP_2_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>U7_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_2_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>APP_2_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_2_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>g_in_a</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_1_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_1_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_1_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>)))
<BR><FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_2_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>U7_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_2_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U7_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_2_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_2_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>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>W</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>g_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_in_a</font><BR>
<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_aa</font><BR>
<FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_aa</font><BR>
<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_out_aaa</font><BR>
<FONT COLOR=#0000cc>U7_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>U7_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_out_a</font><BR>
<FONT COLOR=#0000cc>APP_1_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_1_IN_AAA</font><BR>
<FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_A</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>G_IN_A</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>G_IN_A</font><BR>
<FONT COLOR=#0000cc>U7_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>U7_AAA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>EQ_IN_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>EQ_IN_AA</font><BR>
<FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_A</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_A</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_A</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>APP_2_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_2_IN_AAA</font><BR>
<FONT COLOR=#0000cc>U5_A</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_A</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 2 SCCs with 12 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><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APP_2_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_2_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>g_in_a</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_1_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_1_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_1_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>)))
<BR><FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_2_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>U7_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_2_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U7_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_2_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_2_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>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>W</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>g_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_in_a</font><BR>
<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_aa</font><BR>
<FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_aa</font><BR>
<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_out_aaa</font><BR>
<FONT COLOR=#0000cc>U7_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>U7_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_out_a</font><BR>
<FONT COLOR=#0000cc>APP_2_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_2_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 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_2_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_2_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>APP_2_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_2_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 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_2_IN_AAA</font> &#8594; <FONT COLOR=#0000cc>APP_2_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_2_IN_AAA</font> &#8594; <FONT COLOR=#0000cc>APP_2_IN_AAA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:none<BR><BR><BR>s = <FONT COLOR=#0000cc>APP_2_IN_AAA</font> evaluates to  t =<FONT COLOR=#0000cc>APP_2_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_2_IN_AAA to APP_2_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 <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>APP_1_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_1_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>g_in_a</font>(<FONT COLOR=#cc0000>W</font>) &#8594; <FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>a</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>))))
<BR><FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>S</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>c</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>)))) &#8594; <FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>))
<BR><FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_1_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_1_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_1_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Z</font>)) &#8594; <FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>)))
<BR><FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>Z</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>V</font>))) &#8594; <FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>))
<BR><FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>app_2_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>U7_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_2_in_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>))
<BR><FONT COLOR=#0000cc>U7_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_2_out_aaa</font>(<FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ys</font>, <FONT COLOR=#cc0000>Zs</font>)) &#8594; <FONT COLOR=#0000cc>app_2_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>U5_a</font>(<FONT COLOR=#cc0000>W</font>, <FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>U</font>, <FONT COLOR=#cc0000>W</font>)) &#8594; <FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>W</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>g_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_in_a</font><BR>
<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_aa</font><BR>
<FONT COLOR=#0000cc>eq_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_aa</font><BR>
<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_a</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_1_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_1_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_1_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_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_a</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>app_2_in_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_in_aaa</font><BR>
<FONT COLOR=#0000cc>app_2_out_aaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>app_2_out_aaa</font><BR>
<FONT COLOR=#0000cc>U7_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>U7_aaa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>g_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>g_out_a</font><BR>
<FONT COLOR=#0000cc>APP_1_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_1_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 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>APP_1_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_1_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>APP_1_IN_AAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>APP_1_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 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 NonTerminationProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>APP_1_IN_AAA</font> &#8594; <FONT COLOR=#0000cc>APP_1_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_1_IN_AAA</font> &#8594; <FONT COLOR=#0000cc>APP_1_IN_AAA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:none<BR><BR><BR>s = <FONT COLOR=#0000cc>APP_1_IN_AAA</font> evaluates to  t =<FONT COLOR=#0000cc>APP_1_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_1_IN_AAA to APP_1_IN_AAA.<BR><BR><BR><BR><BR><BR></body>

