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<BR><B>Left Termination</B> of the query pattern
dcsolve_in_2(g, 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 UndefinedPredicateHandlerProof</pre><BR>Clauses:<BR><BR>dcsolve(X, Y)&#160;:-&#160;','(base(X), conquer(X, Y)).<BR>dcsolve(X, Y)&#160;:-&#160;','(divide(X, X0, X1, X2), ','(dcsolve(X1, Y1), ','(dcsolve(X2, Y2), merge(X0, Y1, Y2, Y)))).<BR><BR>Queries:<BR><BR>dcsolve(g,a).<BR><BR>Added facts for all undefined predicates.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><pre>    &#8627 <B>Prolog</B></pre><pre>      &#8627 PrologToPiTRSProof</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>dcsolve(X, Y)&#160;:-&#160;','(base(X), conquer(X, Y)).<BR>dcsolve(X, Y)&#160;:-&#160;','(divide(X, X0, X1, X2), ','(dcsolve(X1, Y1), ','(dcsolve(X2, Y2), merge(X0, Y1, Y2, Y)))).<BR>base(X0).<BR>conquer(X0, X1).<BR>divide(X0, X1, X2, X3).<BR>merge(X0, X1, X2, X3).<BR><BR>Queries:<BR><BR>dcsolve(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>dcsolve_in</font>: (b,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>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_g</font><BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_ga</font><BR>
<FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_ga</font><BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_gaaa</font><BR>
<FONT COLOR=#0000cc>U4_ga</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_aa</font><BR>
<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_a</font><BR>
<FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_a</font><BR>
<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_aa</font><BR>
<FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_aa</font><BR>
<FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_aa</font><BR>
<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_aaaa</font><BR>
<FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U4_aa</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_aa</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>U5_aa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_aaaa</font><BR>
<FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x3</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 UndefinedPredicateHandlerProof</pre><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>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_g</font><BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_ga</font><BR>
<FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_ga</font><BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_gaaa</font><BR>
<FONT COLOR=#0000cc>U4_ga</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_aa</font><BR>
<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_a</font><BR>
<FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_a</font><BR>
<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_aa</font><BR>
<FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_aa</font><BR>
<FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_aa</font><BR>
<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_aaaa</font><BR>
<FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U4_aa</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_aa</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>U5_aa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_aaaa</font><BR>
<FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x3</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>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>BASE_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>CONQUER_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>DIVIDE_IN_GAAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>BASE_IN_A</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>CONQUER_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>DIVIDE_IN_AAAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)
<BR><FONT COLOR=#0000cc>U5_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_AAAA</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)
<BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_AAAA</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_g</font><BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_ga</font><BR>
<FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_ga</font><BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_gaaa</font><BR>
<FONT COLOR=#0000cc>U4_ga</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_aa</font><BR>
<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_a</font><BR>
<FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_a</font><BR>
<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_aa</font><BR>
<FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_aa</font><BR>
<FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_aa</font><BR>
<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_aaaa</font><BR>
<FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U4_aa</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_aa</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>U5_aa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_aaaa</font><BR>
<FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>BASE_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>BASE_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>DIVIDE_IN_AAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>DIVIDE_IN_AAAA</font><BR>
<FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</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>U5_GA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_AA</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>CONQUER_IN_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>CONQUER_IN_AA</font><BR>
<FONT COLOR=#0000cc>BASE_IN_A</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>BASE_IN_A</font><BR>
<FONT COLOR=#0000cc>MERGE_IN_AAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>MERGE_IN_AAAA</font><BR>
<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>DIVIDE_IN_GAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>DIVIDE_IN_GAAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_AA</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>U5_AA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>CONQUER_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>CONQUER_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>DCSOLVE_IN_AA</font><BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><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>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>BASE_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>CONQUER_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>DIVIDE_IN_GAAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>BASE_IN_A</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>CONQUER_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>DIVIDE_IN_AAAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)
<BR><FONT COLOR=#0000cc>U5_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_AAAA</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)
<BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_AAAA</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_g</font><BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_ga</font><BR>
<FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_ga</font><BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_gaaa</font><BR>
<FONT COLOR=#0000cc>U4_ga</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_aa</font><BR>
<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_a</font><BR>
<FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_a</font><BR>
<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_aa</font><BR>
<FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_aa</font><BR>
<FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_aa</font><BR>
<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_aaaa</font><BR>
<FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U4_aa</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_aa</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>U5_aa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_aaaa</font><BR>
<FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>BASE_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>BASE_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>DIVIDE_IN_AAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>DIVIDE_IN_AAAA</font><BR>
<FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</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>U5_GA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_AA</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>CONQUER_IN_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>CONQUER_IN_AA</font><BR>
<FONT COLOR=#0000cc>BASE_IN_A</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>BASE_IN_A</font><BR>
<FONT COLOR=#0000cc>MERGE_IN_AAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>MERGE_IN_AAAA</font><BR>
<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>DIVIDE_IN_GAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>DIVIDE_IN_GAAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_AA</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>U5_AA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>CONQUER_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>CONQUER_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>DCSOLVE_IN_AA</font><BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 1 SCC with 20 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</pre><pre>        &#8627 PiTRS</pre><pre>          &#8627 DependencyPairsProof</pre><pre>            &#8627 PiDP</pre><pre>              &#8627 DependencyGraphProof</pre><pre>                &#8627 <B>PiDP</B></pre><pre>                  &#8627 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>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_g</font><BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_ga</font><BR>
<FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_ga</font><BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_gaaa</font><BR>
<FONT COLOR=#0000cc>U4_ga</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_aa</font><BR>
<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_a</font><BR>
<FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_a</font><BR>
<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_aa</font><BR>
<FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_aa</font><BR>
<FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_aa</font><BR>
<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_aaaa</font><BR>
<FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U4_aa</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_aa</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>U5_aa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_aaaa</font><BR>
<FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_AA</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>DCSOLVE_IN_AA</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 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</pre><pre>        &#8627 PiTRS</pre><pre>          &#8627 DependencyPairsProof</pre><pre>            &#8627 PiDP</pre><pre>              &#8627 DependencyGraphProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 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>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_aa</font><BR>
<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_a</font><BR>
<FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_a</font><BR>
<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_aa</font><BR>
<FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_aa</font><BR>
<FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_aa</font><BR>
<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_aaaa</font><BR>
<FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U4_aa</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_aa</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>U5_aa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_aaaa</font><BR>
<FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_AA</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>DCSOLVE_IN_AA</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 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</pre><pre>        &#8627 PiTRS</pre><pre>          &#8627 DependencyPairsProof</pre><pre>            &#8627 PiDP</pre><pre>              &#8627 DependencyGraphProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 UsableRulesProof</pre><pre>                    &#8627 PiDP</pre><pre>                      &#8627 PiDPToQDPProof</pre><pre>                        &#8627 <B>QDP</B></pre><pre>                          &#8627 Rewriting</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font> &#8594; <FONT COLOR=#0000cc>base_out_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font> &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font></BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>) at position [0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</pre><pre>        &#8627 PiTRS</pre><pre>          &#8627 DependencyPairsProof</pre><pre>            &#8627 PiDP</pre><pre>              &#8627 DependencyGraphProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 UsableRulesProof</pre><pre>                    &#8627 PiDP</pre><pre>                      &#8627 PiDPToQDPProof</pre><pre>                        &#8627 QDP</pre><pre>                          &#8627 Rewriting</pre><pre>                            &#8627 <B>QDP</B></pre><pre>                              &#8627 Narrowing</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font> &#8594; <FONT COLOR=#0000cc>base_out_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font> &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font></BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By narrowing [15] the rule <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>) at position [0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>))
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</pre><pre>        &#8627 PiTRS</pre><pre>          &#8627 DependencyPairsProof</pre><pre>            &#8627 PiDP</pre><pre>              &#8627 DependencyGraphProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 UsableRulesProof</pre><pre>                    &#8627 PiDP</pre><pre>                      &#8627 PiDPToQDPProof</pre><pre>                        &#8627 QDP</pre><pre>                          &#8627 Rewriting</pre><pre>                            &#8627 QDP</pre><pre>                              &#8627 Narrowing</pre><pre>                                &#8627 <B>QDP</B></pre><pre>                                  &#8627 Rewriting</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>))
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>))
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font> &#8594; <FONT COLOR=#0000cc>base_out_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font> &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font></BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)) at position [0,0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</pre><pre>        &#8627 PiTRS</pre><pre>          &#8627 DependencyPairsProof</pre><pre>            &#8627 PiDP</pre><pre>              &#8627 DependencyGraphProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 UsableRulesProof</pre><pre>                    &#8627 PiDP</pre><pre>                      &#8627 PiDPToQDPProof</pre><pre>                        &#8627 QDP</pre><pre>                          &#8627 Rewriting</pre><pre>                            &#8627 QDP</pre><pre>                              &#8627 Narrowing</pre><pre>                                &#8627 QDP</pre><pre>                                  &#8627 Rewriting</pre><pre>                                    &#8627 <B>QDP</B></pre><pre>                                      &#8627 Rewriting</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>))
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font> &#8594; <FONT COLOR=#0000cc>base_out_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font> &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font></BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)) at position [0,0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</pre><pre>        &#8627 PiTRS</pre><pre>          &#8627 DependencyPairsProof</pre><pre>            &#8627 PiDP</pre><pre>              &#8627 DependencyGraphProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 UsableRulesProof</pre><pre>                    &#8627 PiDP</pre><pre>                      &#8627 PiDPToQDPProof</pre><pre>                        &#8627 QDP</pre><pre>                          &#8627 Rewriting</pre><pre>                            &#8627 QDP</pre><pre>                              &#8627 Narrowing</pre><pre>                                &#8627 QDP</pre><pre>                                  &#8627 Rewriting</pre><pre>                                    &#8627 QDP</pre><pre>                                      &#8627 Rewriting</pre><pre>                                        &#8627 <B>QDP</B></pre><pre>                                          &#8627 Rewriting</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>))
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>))
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font> &#8594; <FONT COLOR=#0000cc>base_out_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font> &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font></BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>)) at position [0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</pre><pre>        &#8627 PiTRS</pre><pre>          &#8627 DependencyPairsProof</pre><pre>            &#8627 PiDP</pre><pre>              &#8627 DependencyGraphProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 UsableRulesProof</pre><pre>                    &#8627 PiDP</pre><pre>                      &#8627 PiDPToQDPProof</pre><pre>                        &#8627 QDP</pre><pre>                          &#8627 Rewriting</pre><pre>                            &#8627 QDP</pre><pre>                              &#8627 Narrowing</pre><pre>                                &#8627 QDP</pre><pre>                                  &#8627 Rewriting</pre><pre>                                    &#8627 QDP</pre><pre>                                      &#8627 Rewriting</pre><pre>                                        &#8627 QDP</pre><pre>                                          &#8627 Rewriting</pre><pre>                                            &#8627 <B>QDP</B></pre><pre>                                              &#8627 Rewriting</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>))
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font> &#8594; <FONT COLOR=#0000cc>base_out_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font> &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font></BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)) at position [0,0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</pre><pre>        &#8627 PiTRS</pre><pre>          &#8627 DependencyPairsProof</pre><pre>            &#8627 PiDP</pre><pre>              &#8627 DependencyGraphProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 UsableRulesProof</pre><pre>                    &#8627 PiDP</pre><pre>                      &#8627 PiDPToQDPProof</pre><pre>                        &#8627 QDP</pre><pre>                          &#8627 Rewriting</pre><pre>                            &#8627 QDP</pre><pre>                              &#8627 Narrowing</pre><pre>                                &#8627 QDP</pre><pre>                                  &#8627 Rewriting</pre><pre>                                    &#8627 QDP</pre><pre>                                      &#8627 Rewriting</pre><pre>                                        &#8627 QDP</pre><pre>                                          &#8627 Rewriting</pre><pre>                                            &#8627 QDP</pre><pre>                                              &#8627 Rewriting</pre><pre>                                                &#8627 <B>QDP</B></pre><pre>                                                  &#8627 Rewriting</pre><pre>      &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>))
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>))
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font> &#8594; <FONT COLOR=#0000cc>base_out_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font> &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font></BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>)) at position [0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><pre>    &#8627 Prolog</pre><pre>      &#8627 PrologToPiTRSProof</pre><pre>        &#8627 PiTRS</pre><pre>          &#8627 DependencyPairsProof</pre><pre>            &#8627 PiDP</pre><pre>              &#8627 DependencyGraphProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 UsableRulesProof</pre><pre>                    &#8627 PiDP</pre><pre>                      &#8627 PiDPToQDPProof</pre><pre>                        &#8627 QDP</pre><pre>                          &#8627 Rewriting</pre><pre>                            &#8627 QDP</pre><pre>                              &#8627 Narrowing</pre><pre>                                &#8627 QDP</pre><pre>                                  &#8627 Rewriting</pre><pre>                                    &#8627 QDP</pre><pre>                                      &#8627 Rewriting</pre><pre>                                        &#8627 QDP</pre><pre>                                          &#8627 Rewriting</pre><pre>                                            &#8627 QDP</pre><pre>                                              &#8627 Rewriting</pre><pre>                                                &#8627 QDP</pre><pre>                                                  &#8627 Rewriting</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>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font> &#8594; <FONT COLOR=#0000cc>base_out_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font> &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font></BLOCKQUOTE><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 narrowing to the left:<BR><BR>The TRS P consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font> &#8594; <FONT COLOR=#0000cc>base_out_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font> &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font></BLOCKQUOTE><BR><BR>s = <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) evaluates to  t =<FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</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><B><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)</B> &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font><BR>with rule <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> at position [] and matcher [ ]<BR><BR><B><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></B> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)<BR>with rule <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)<BR><BR>Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence<BR><BR><BR>All these steps are and every following step will be a correct step w.r.t to Q.<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>dcsolve_in</font>: (b,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>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_aa</font><BR>
<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_a</font><BR>
<FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_a</font><BR>
<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_aa</font><BR>
<FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_aa</font><BR>
<FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_aa</font><BR>
<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_aaaa</font><BR>
<FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U4_aa</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_aa</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>U5_aa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_aaaa</font><BR>
<FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</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 UndefinedPredicateHandlerProof</pre><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>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_aa</font><BR>
<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_a</font><BR>
<FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_a</font><BR>
<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_aa</font><BR>
<FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_aa</font><BR>
<FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_aa</font><BR>
<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_aaaa</font><BR>
<FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U4_aa</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_aa</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>U5_aa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_aaaa</font><BR>
<FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</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>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>BASE_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>CONQUER_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>DIVIDE_IN_GAAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>BASE_IN_A</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>CONQUER_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>DIVIDE_IN_AAAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)
<BR><FONT COLOR=#0000cc>U5_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_AAAA</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)
<BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_AAAA</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_aa</font><BR>
<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_a</font><BR>
<FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_a</font><BR>
<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_aa</font><BR>
<FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_aa</font><BR>
<FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_aa</font><BR>
<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_aaaa</font><BR>
<FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U4_aa</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_aa</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>U5_aa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_aaaa</font><BR>
<FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>BASE_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>BASE_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>DIVIDE_IN_AAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>DIVIDE_IN_AAAA</font><BR>
<FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</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>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_AA</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>CONQUER_IN_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>CONQUER_IN_AA</font><BR>
<FONT COLOR=#0000cc>BASE_IN_A</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>BASE_IN_A</font><BR>
<FONT COLOR=#0000cc>MERGE_IN_AAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>MERGE_IN_AAAA</font><BR>
<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>DIVIDE_IN_GAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>DIVIDE_IN_GAAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_AA</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>U5_AA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>CONQUER_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>CONQUER_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>DCSOLVE_IN_AA</font><BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><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>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>BASE_IN_G</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>CONQUER_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>DIVIDE_IN_GAAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>BASE_IN_A</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>CONQUER_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>DIVIDE_IN_AAAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)
<BR><FONT COLOR=#0000cc>U5_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_AAAA</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)
<BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>MERGE_IN_AAAA</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_aa</font><BR>
<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_a</font><BR>
<FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_a</font><BR>
<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_aa</font><BR>
<FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_aa</font><BR>
<FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_aa</font><BR>
<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_aaaa</font><BR>
<FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U4_aa</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_aa</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>U5_aa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_aaaa</font><BR>
<FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>BASE_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>BASE_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>DIVIDE_IN_AAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>DIVIDE_IN_AAAA</font><BR>
<FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</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>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_AA</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>CONQUER_IN_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>CONQUER_IN_AA</font><BR>
<FONT COLOR=#0000cc>BASE_IN_A</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>BASE_IN_A</font><BR>
<FONT COLOR=#0000cc>MERGE_IN_AAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>MERGE_IN_AAAA</font><BR>
<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>DIVIDE_IN_GAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>DIVIDE_IN_GAAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_AA</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>U5_AA</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>DCSOLVE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>CONQUER_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>CONQUER_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_GA</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>DCSOLVE_IN_AA</font><BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 1 SCC with 20 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><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 <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>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U4_ga</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_aa</font><BR>
<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_a</font><BR>
<FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_a</font><BR>
<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_aa</font><BR>
<FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_aa</font><BR>
<FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_aa</font><BR>
<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_aaaa</font><BR>
<FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U4_aa</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_aa</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>U5_aa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_aaaa</font><BR>
<FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U5_ga</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>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_AA</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>DCSOLVE_IN_AA</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 UndefinedPredicateHandlerProof</pre><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 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>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>) &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>))
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>)) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>))
<BR><FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>X0</font>) &#8594; <FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>X0</font>)
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>Y1</font>)) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>))
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>) &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>)
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>Y2</font>)) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>Y1</font>, <FONT COLOR=#cc0000>Y2</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>) &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>X0</font>, <FONT COLOR=#cc0000>X1</font>, <FONT COLOR=#cc0000>X2</font>, <FONT COLOR=#cc0000>X3</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_in_aa</font><BR>
<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>base_in_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_in_a</font><BR>
<FONT COLOR=#0000cc>base_out_a</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>base_out_a</font><BR>
<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>conquer_in_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_in_aa</font><BR>
<FONT COLOR=#0000cc>conquer_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>conquer_out_aa</font><BR>
<FONT COLOR=#0000cc>dcsolve_out_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>dcsolve_out_aa</font><BR>
<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>divide_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_in_aaaa</font><BR>
<FONT COLOR=#0000cc>divide_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>divide_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U4_aa</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U5_aa</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>U5_aa</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>merge_in_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_in_aaaa</font><BR>
<FONT COLOR=#0000cc>merge_out_aaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>merge_out_aaaa</font><BR>
<FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_AA</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>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>DCSOLVE_IN_AA</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 UndefinedPredicateHandlerProof</pre><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 PiDP</pre><pre>                  &#8627 UsableRulesProof</pre><pre>                    &#8627 PiDP</pre><pre>                      &#8627 PiDPToQDPProof</pre><pre>                        &#8627 <B>QDP</B></pre><pre>                          &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font> &#8594; <FONT COLOR=#0000cc>base_out_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font> &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font></BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>) at position [0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><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 PiDP</pre><pre>                  &#8627 UsableRulesProof</pre><pre>                    &#8627 PiDP</pre><pre>                      &#8627 PiDPToQDPProof</pre><pre>                        &#8627 QDP</pre><pre>                          &#8627 Rewriting</pre><pre>                            &#8627 <B>QDP</B></pre><pre>                              &#8627 Narrowing</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font> &#8594; <FONT COLOR=#0000cc>base_out_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font> &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font></BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By narrowing [15] the rule <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>) at position [0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>))
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><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 PiDP</pre><pre>                  &#8627 UsableRulesProof</pre><pre>                    &#8627 PiDP</pre><pre>                      &#8627 PiDPToQDPProof</pre><pre>                        &#8627 QDP</pre><pre>                          &#8627 Rewriting</pre><pre>                            &#8627 QDP</pre><pre>                              &#8627 Narrowing</pre><pre>                                &#8627 <B>QDP</B></pre><pre>                                  &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>))
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>))
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font> &#8594; <FONT COLOR=#0000cc>base_out_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font> &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font></BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)) at position [0,0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><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 PiDP</pre><pre>                  &#8627 UsableRulesProof</pre><pre>                    &#8627 PiDP</pre><pre>                      &#8627 PiDPToQDPProof</pre><pre>                        &#8627 QDP</pre><pre>                          &#8627 Rewriting</pre><pre>                            &#8627 QDP</pre><pre>                              &#8627 Narrowing</pre><pre>                                &#8627 QDP</pre><pre>                                  &#8627 Rewriting</pre><pre>                                    &#8627 <B>QDP</B></pre><pre>                                      &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>))
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font> &#8594; <FONT COLOR=#0000cc>base_out_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font> &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font></BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)) at position [0,0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><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 PiDP</pre><pre>                  &#8627 UsableRulesProof</pre><pre>                    &#8627 PiDP</pre><pre>                      &#8627 PiDPToQDPProof</pre><pre>                        &#8627 QDP</pre><pre>                          &#8627 Rewriting</pre><pre>                            &#8627 QDP</pre><pre>                              &#8627 Narrowing</pre><pre>                                &#8627 QDP</pre><pre>                                  &#8627 Rewriting</pre><pre>                                    &#8627 QDP</pre><pre>                                      &#8627 Rewriting</pre><pre>                                        &#8627 <B>QDP</B></pre><pre>                                          &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>))
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>))
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font> &#8594; <FONT COLOR=#0000cc>base_out_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font> &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font></BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>)) at position [0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><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 PiDP</pre><pre>                  &#8627 UsableRulesProof</pre><pre>                    &#8627 PiDP</pre><pre>                      &#8627 PiDPToQDPProof</pre><pre>                        &#8627 QDP</pre><pre>                          &#8627 Rewriting</pre><pre>                            &#8627 QDP</pre><pre>                              &#8627 Narrowing</pre><pre>                                &#8627 QDP</pre><pre>                                  &#8627 Rewriting</pre><pre>                                    &#8627 QDP</pre><pre>                                      &#8627 Rewriting</pre><pre>                                        &#8627 QDP</pre><pre>                                          &#8627 Rewriting</pre><pre>                                            &#8627 <B>QDP</B></pre><pre>                                              &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>))
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font> &#8594; <FONT COLOR=#0000cc>base_out_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font> &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font></BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)) at position [0,0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><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 PiDP</pre><pre>                  &#8627 UsableRulesProof</pre><pre>                    &#8627 PiDP</pre><pre>                      &#8627 PiDPToQDPProof</pre><pre>                        &#8627 QDP</pre><pre>                          &#8627 Rewriting</pre><pre>                            &#8627 QDP</pre><pre>                              &#8627 Narrowing</pre><pre>                                &#8627 QDP</pre><pre>                                  &#8627 Rewriting</pre><pre>                                    &#8627 QDP</pre><pre>                                      &#8627 Rewriting</pre><pre>                                        &#8627 QDP</pre><pre>                                          &#8627 Rewriting</pre><pre>                                            &#8627 QDP</pre><pre>                                              &#8627 Rewriting</pre><pre>                                                &#8627 <B>QDP</B></pre><pre>                                                  &#8627 Rewriting</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>))
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>))
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font> &#8594; <FONT COLOR=#0000cc>base_out_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font> &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font></BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By rewriting [15] the rule <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>)) at position [0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 UndefinedPredicateHandlerProof</pre><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 PiDP</pre><pre>                  &#8627 UsableRulesProof</pre><pre>                    &#8627 PiDP</pre><pre>                      &#8627 PiDPToQDPProof</pre><pre>                        &#8627 QDP</pre><pre>                          &#8627 Rewriting</pre><pre>                            &#8627 QDP</pre><pre>                              &#8627 Narrowing</pre><pre>                                &#8627 QDP</pre><pre>                                  &#8627 Rewriting</pre><pre>                                    &#8627 QDP</pre><pre>                                      &#8627 Rewriting</pre><pre>                                        &#8627 QDP</pre><pre>                                          &#8627 Rewriting</pre><pre>                                            &#8627 QDP</pre><pre>                                              &#8627 Rewriting</pre><pre>                                                &#8627 QDP</pre><pre>                                                  &#8627 Rewriting</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>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font> &#8594; <FONT COLOR=#0000cc>base_out_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font> &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font>
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font></BLOCKQUOTE><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 narrowing to the left:<BR><BR>The TRS P consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>))
<BR><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)
<BR><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font>
<BR><FONT COLOR=#0000cc>U4_AA</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>divide_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>divide_out_aaaa</font>
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_in_a</font>)
<BR><FONT COLOR=#0000cc>dcsolve_in_aa</font> &#8594; <FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U1_aa</font>(<FONT COLOR=#0000cc>base_out_a</font>) &#8594; <FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_in_aa</font>)
<BR><FONT COLOR=#0000cc>U3_aa</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>base_in_a</font> &#8594; <FONT COLOR=#0000cc>base_out_a</font>
<BR><FONT COLOR=#0000cc>U2_aa</font>(<FONT COLOR=#0000cc>conquer_out_aa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>U4_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_in_aa</font>)
<BR><FONT COLOR=#0000cc>conquer_in_aa</font> &#8594; <FONT COLOR=#0000cc>conquer_out_aa</font>
<BR><FONT COLOR=#0000cc>U5_aa</font>(<FONT COLOR=#0000cc>dcsolve_out_aa</font>) &#8594; <FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_in_aaaa</font>)
<BR><FONT COLOR=#0000cc>U6_aa</font>(<FONT COLOR=#0000cc>merge_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>dcsolve_out_aa</font>
<BR><FONT COLOR=#0000cc>merge_in_aaaa</font> &#8594; <FONT COLOR=#0000cc>merge_out_aaaa</font></BLOCKQUOTE><BR><BR>s = <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) evaluates to  t =<FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</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><B><FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)</B> &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font><BR>with rule <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>) &#8594; <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> at position [] and matcher [ ]<BR><BR><B><FONT COLOR=#0000cc>DCSOLVE_IN_AA</font></B> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)<BR>with rule <FONT COLOR=#0000cc>DCSOLVE_IN_AA</font> &#8594; <FONT COLOR=#0000cc>U3_AA</font>(<FONT COLOR=#0000cc>divide_out_aaaa</font>)<BR><BR>Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence<BR><BR><BR>All these steps are and every following step will be a correct step w.r.t to Q.<BR><BR><BR><BR><BR><BR></body>

