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<title>Left Termination proof of ../tpdb/LP/BCGGV05/insert-bff.pl</title>
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<BR><B>Left Termination</B> of the query pattern
insert_in_3(g, a, a)
w.r.t. the given <I>Prolog program</I> could not be shown:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>insert(X, void, tree(X, void, void)).<BR>insert(X, tree(X, Left, Right), tree(X, Left, Right)).<BR>insert(X, tree(Y, Left, Right), tree(Y, Left1, Right))&#160;:-&#160;','(less(X, Y), insert(X, Left, Left1)).<BR>insert(X, tree(Y, Left, Right), tree(Y, Left, Right1))&#160;:-&#160;','(less(Y, X), insert(X, Right, Right1)).<BR>less(0, s(X)).<BR>less(s(X), s(Y))&#160;:-&#160;less(X, Y).<BR><BR>Queries:<BR><BR>insert(g,a,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>insert_in</font>: (b,f,f)
<BR><FONT COLOR=#0000cc>less_in</font>: (b,f) (f,b)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</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=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_out_gaa</font><BR>
<FONT COLOR=#0000cc>U1_gaa</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>U1_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ga</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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>U3_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 <B>PiTRS</B></pre><pre>      &#8627 DependencyPairsProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</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=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_out_gaa</font><BR>
<FONT COLOR=#0000cc>U1_gaa</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>U1_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ga</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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>U3_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x6</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>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</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=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_out_gaa</font><BR>
<FONT COLOR=#0000cc>U1_gaa</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>U1_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ga</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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>U3_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_GAA</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>U2_GAA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_GAA</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_GAA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_GAA</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>U3_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_GAA</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>U1_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 <B>PiDP</B></pre><pre>          &#8627 DependencyGraphProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</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=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_out_gaa</font><BR>
<FONT COLOR=#0000cc>U1_gaa</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>U1_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ga</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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>U3_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_GAA</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>U2_GAA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_GAA</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_GAA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_GAA</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>U3_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_GAA</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>U1_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 3 SCCs with 6 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</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=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_out_gaa</font><BR>
<FONT COLOR=#0000cc>U1_gaa</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>U1_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ga</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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>U3_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</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=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_out_gaa</font><BR>
<FONT COLOR=#0000cc>U1_gaa</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>U1_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ga</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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>U3_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>)<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</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=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_out_gaa</font><BR>
<FONT COLOR=#0000cc>U1_gaa</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>U1_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ga</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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>U3_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_GAA</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>U3_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U1_GAA</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>U1_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</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=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ga</font><BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_GAA</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>U3_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U1_GAA</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>U1_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 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>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By narrowing [15] the rule <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>)) at position [1] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</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>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By narrowing [15] the rule <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>)) at position [1] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Narrowing</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 Instantiation</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Narrowing</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Instantiation</pre><pre>                                  &#8627 <B>QDP</B></pre><pre>                                    &#8627 DependencyGraphProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Narrowing</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Instantiation</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 DependencyGraphProof</pre><pre>                                      &#8627 AND</pre><pre>                                        &#8627 <B>QDP</B></pre><pre>                                          &#8627 UsableRulesProof</pre><pre>                                        &#8627 QDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Narrowing</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Instantiation</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 DependencyGraphProof</pre><pre>                                      &#8627 AND</pre><pre>                                        &#8627 QDP</pre><pre>                                          &#8627 UsableRulesProof</pre><pre>                                            &#8627 <B>QDP</B></pre><pre>                                              &#8627 QReductionProof</pre><pre>                                        &#8627 QDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>)</BLOCKQUOTE><BR>R is empty.<BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Narrowing</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Instantiation</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 DependencyGraphProof</pre><pre>                                      &#8627 AND</pre><pre>                                        &#8627 QDP</pre><pre>                                          &#8627 UsableRulesProof</pre><pre>                                            &#8627 QDP</pre><pre>                                              &#8627 QReductionProof</pre><pre>                                                &#8627 <B>QDP</B></pre><pre>                                                  &#8627 NonTerminationProof</pre><pre>                                        &#8627 QDP</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>We used the non-termination processor [17] to show that the DP problem is infinite.<BR>Found a loop by narrowing to the left:<BR><BR>The TRS P consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:none<BR><BR><BR>s = <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) evaluates to  t =<FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</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>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>)</B> &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>)<BR>with rule <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>) at position [] and matcher [ ]<BR><BR><B><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>)</B> &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>)<BR>with rule <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</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><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Narrowing</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Instantiation</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 DependencyGraphProof</pre><pre>                                      &#8627 AND</pre><pre>                                        &#8627 QDP</pre><pre>                                        &#8627 <B>QDP</B></pre><pre>                                          &#8627 Instantiation</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Narrowing</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Instantiation</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 DependencyGraphProof</pre><pre>                                      &#8627 AND</pre><pre>                                        &#8627 QDP</pre><pre>                                        &#8627 QDP</pre><pre>                                          &#8627 Instantiation</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>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</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>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#0000cc>less_out_ga</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR><BR>s = <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>)) evaluates to  t =<FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</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_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>))</B> &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))<BR>with rule <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0'</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0'</font>)) at position [] and matcher [<FONT COLOR=#cc0000>z0'</font> / <FONT COLOR=#cc0000>z0</font>]<BR><BR><B><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))</B> &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>))<BR>with rule <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</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>insert_in</font>: (b,f,f)
<BR><FONT COLOR=#0000cc>less_in</font>: (b,f) (f,b)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</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=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gaa</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>U1_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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>U3_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 <B>PiTRS</B></pre><pre>      &#8627 DependencyPairsProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</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=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gaa</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>U1_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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>U3_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</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>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</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=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gaa</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>U1_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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>U3_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_GAA</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>U2_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_GAA</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_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_GAA</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>U3_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_GAA</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>U1_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 <B>PiDP</B></pre><pre>          &#8627 DependencyGraphProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</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=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gaa</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>U1_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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>U3_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_GAA</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>U2_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_GAA</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_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_GAA</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>U3_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_GAA</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>U1_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 3 SCCs with 6 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</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=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gaa</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>U1_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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>U3_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_AG</font>(<FONT COLOR=#cc0000>Y</font>)<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</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=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gaa</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>U1_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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>U3_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>LESS_IN_GA</font>(<FONT COLOR=#cc0000>X</font>)<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>void</font>, <FONT COLOR=#0000cc>void</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</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=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>))
<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U3_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U4_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>))
<BR><FONT COLOR=#0000cc>U2_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)) &#8594; <FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>insert_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_gaa</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>U1_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_gaa</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>U2_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gaa</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>U3_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U4_gaa</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_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_GAA</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>U3_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U1_GAA</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>U1_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#cc0000>Right</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Left1</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Left1</font>)
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right1</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Left</font>, <FONT COLOR=#cc0000>Right</font>, <FONT COLOR=#cc0000>Right1</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</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=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>0</font>&#160; = &#160;<FONT COLOR=#0000cc>0</font><BR>
<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U3_GAA</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>U3_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U1_GAA</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>U1_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 Narrowing</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By narrowing [15] the rule <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>)) at position [1] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</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>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By narrowing [15] the rule <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>X</font>)) at position [1] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Narrowing</pre><pre>                              &#8627 <B>QDP</B></pre><pre>                                &#8627 Instantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Narrowing</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Instantiation</pre><pre>                                  &#8627 <B>QDP</B></pre><pre>                                    &#8627 DependencyGraphProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Narrowing</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Instantiation</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 DependencyGraphProof</pre><pre>                                      &#8627 AND</pre><pre>                                        &#8627 <B>QDP</B></pre><pre>                                          &#8627 UsableRulesProof</pre><pre>                                        &#8627 QDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Narrowing</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Instantiation</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 DependencyGraphProof</pre><pre>                                      &#8627 AND</pre><pre>                                        &#8627 QDP</pre><pre>                                          &#8627 UsableRulesProof</pre><pre>                                            &#8627 <B>QDP</B></pre><pre>                                              &#8627 QReductionProof</pre><pre>                                        &#8627 QDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>))</BLOCKQUOTE><BR>R is empty.<BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Narrowing</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Instantiation</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 DependencyGraphProof</pre><pre>                                      &#8627 AND</pre><pre>                                        &#8627 QDP</pre><pre>                                          &#8627 UsableRulesProof</pre><pre>                                            &#8627 QDP</pre><pre>                                              &#8627 QReductionProof</pre><pre>                                                &#8627 <B>QDP</B></pre><pre>                                                  &#8627 NonTerminationProof</pre><pre>                                        &#8627 QDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>))</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>We used the non-termination processor [17] to show that the DP problem is infinite.<BR>Found a loop by narrowing to the left:<BR><BR>The TRS P consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:none<BR><BR><BR>s = <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) evaluates to  t =<FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</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>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>)</B> &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>))<BR>with rule <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>)) at position [] and matcher [ ]<BR><BR><B><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>))</B> &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</font>)<BR>with rule <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>0</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><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Narrowing</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Instantiation</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 DependencyGraphProof</pre><pre>                                      &#8627 AND</pre><pre>                                        &#8627 QDP</pre><pre>                                        &#8627 <B>QDP</B></pre><pre>                                          &#8627 Instantiation</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By instantiating [15] the rule <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#cc0000>X</font>) we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 Narrowing</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 Narrowing</pre><pre>                              &#8627 QDP</pre><pre>                                &#8627 Instantiation</pre><pre>                                  &#8627 QDP</pre><pre>                                    &#8627 DependencyGraphProof</pre><pre>                                      &#8627 AND</pre><pre>                                        &#8627 QDP</pre><pre>                                        &#8627 QDP</pre><pre>                                          &#8627 Instantiation</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>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</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>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))
<BR><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)))
<BR><FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))
<BR><FONT COLOR=#0000cc>U1_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>0</font>) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>0</font>)
<BR><FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_in_ga</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_in_ag</font>(<FONT COLOR=#cc0000>Y</font>))
<BR><FONT COLOR=#0000cc>U5_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#cc0000>X</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U5_ag</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>Y</font>)) &#8594; <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>Y</font>))</BLOCKQUOTE><BR><BR>s = <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))) evaluates to  t =<FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</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_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>)))</B> &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))<BR>with rule <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0'</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0'</font>))) &#8594; <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0'</font>)) at position [] and matcher [<FONT COLOR=#cc0000>z0'</font> / <FONT COLOR=#cc0000>z0</font>]<BR><BR><B><FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>))</B> &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>z0</font>)))<BR>with rule <FONT COLOR=#0000cc>INSERT_IN_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>)) &#8594; <FONT COLOR=#0000cc>U3_GAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</font>), <FONT COLOR=#0000cc>less_out_ag</font>(<FONT COLOR=#0000cc>0</font>, <FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>x0</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>

