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<title>Left Termination proof of ../tpdb/LP/SGST06/hbal_tree.pl</title>
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<BR><B>Left Termination</B> of the query pattern
hbal_tree_in_2(g, a)
w.r.t. the given <I>Prolog program</I> could successfully be <font color=#00ff00>proven</font>:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>hbal_tree(zero, nil).<BR>hbal_tree(s(zero), t(x, nil, nil)).<BR>hbal_tree(s(s(X)), t(x, L, R))&#160;:-&#160;','(distr(s(X), X, DL, DR), ','(hbal_tree(DL, L), hbal_tree(DR, R))).<BR>distr(D1, X, D1, D1).<BR>distr(D1, D2, D1, D2).<BR>distr(D1, D2, D2, D1).<BR><BR>Queries:<BR><BR>hbal_tree(g,a).<BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>hbal_tree_in</font>: (b,f)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>))
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D1</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D1</font>)
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>)
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>x2</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>t</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>t</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>x</font>&#160; = &#160;<FONT COLOR=#0000cc>x</font><BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 <B>PiTRS</B></pre><pre>      &#8627 DependencyPairsProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>))
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D1</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D1</font>)
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>)
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>x2</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>t</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>t</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>x</font>&#160; = &#160;<FONT COLOR=#0000cc>x</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>HBAL_TREE_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>))
<BR><FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>DISTR_IN_GGAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)) &#8594; <FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#cc0000>R</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>))
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D1</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D1</font>)
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>)
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>x2</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>t</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>t</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>x</font>&#160; = &#160;<FONT COLOR=#0000cc>x</font><BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>DISTR_IN_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>DISTR_IN_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 <B>PiDP</B></pre><pre>          &#8627 DependencyGraphProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>))
<BR><FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>DISTR_IN_GGAA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)) &#8594; <FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#cc0000>R</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>))
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D1</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D1</font>)
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>)
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>x2</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>t</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>t</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>x</font>&#160; = &#160;<FONT COLOR=#0000cc>x</font><BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GA</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>DISTR_IN_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>DISTR_IN_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 1 SCC with 2 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 <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>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#cc0000>R</font>)
<BR><FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)) &#8594; <FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>nil</font>) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>zero</font>, <FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>)) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>zero</font>), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>))
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D1</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D1</font>)
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>)
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>)), <FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>x2</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>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>t</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>t</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>x</font>&#160; = &#160;<FONT COLOR=#0000cc>x</font><BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 PiDP</pre><pre>              &#8627 PiDPToQDPProof</pre><pre>                &#8627 <B>QDP</B></pre><pre>                  &#8627 QDPOrderProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)) &#8594; <FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#cc0000>DL</font>)
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#cc0000>DR</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DL</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>zero</font>)) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D1</font>)
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>)
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DL</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DR</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We use the reduction pair processor [15].<P><BR>The following pairs can be oriented strictly and are deleted.<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)) &#8594; <FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#cc0000>DL</font>)
<BR><FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#cc0000>DR</font>)
<BR><FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)) &#8594; <FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DL</font>))</BLOCKQUOTE>The remaining pairs can at least be oriented weakly.<BR>none<BR>Used ordering:  Combined order from the following AFS and order.<BR><FONT COLOR=#0000cc>HBAL_TREE_IN_GA</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>HBAL_TREE_IN_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>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_GA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>hbal_tree_out_ga</font><BR>
<FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>hbal_tree_in_ga</font><BR>
<FONT COLOR=#0000cc>zero</font>&#160; = &#160;<FONT COLOR=#0000cc>zero</font><BR>
<FONT COLOR=#0000cc>t</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#cc0000>x2</font><BR>
<FONT COLOR=#0000cc>x</font>&#160; = &#160;<FONT COLOR=#0000cc>x</font><BR>
<FONT COLOR=#0000cc>nil</font>&#160; = &#160;<FONT COLOR=#0000cc>nil</font><BR>
<FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#cc0000>x2</font><BR>
<FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ga</font><BR>
<FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#cc0000>x2</font><BR>
<BR>Recursive path order with status [2].<BR>Quasi-Precedence: <BLOCKQUOTE>
[HBAL<SUB><FONT SIZE="-1">TREE<SUB><FONT SIZE="-1">IN<SUB><FONT SIZE="-1">GA<SUB><FONT SIZE="-1">1</FONT></SUB></FONT></SUB></FONT></SUB></FONT></SUB>, s<SUB><FONT SIZE="-1">1</FONT></SUB>, U1<SUB><FONT SIZE="-1">GA<SUB><FONT SIZE="-1">1</FONT></SUB></FONT></SUB>, U2<SUB><FONT SIZE="-1">GA<SUB><FONT SIZE="-1">2</FONT></SUB></FONT></SUB>, hbal<SUB><FONT SIZE="-1">tree<SUB><FONT SIZE="-1">out<SUB><FONT SIZE="-1">ga</FONT></SUB></FONT></SUB></FONT></SUB>, hbal<SUB><FONT SIZE="-1">tree<SUB><FONT SIZE="-1">in<SUB><FONT SIZE="-1">ga</FONT></SUB></FONT></SUB></FONT></SUB>, zero, U1<SUB><FONT SIZE="-1">ga]</FONT></SUB> &gt; distr<SUB><FONT SIZE="-1">in<SUB><FONT SIZE="-1">ggaa<SUB><FONT SIZE="-1">2</FONT></SUB></FONT></SUB></FONT></SUB> &gt; distr<SUB><FONT SIZE="-1">out<SUB><FONT SIZE="-1">ggaa<SUB><FONT SIZE="-1">2</FONT></SUB></FONT></SUB></FONT></SUB><BR>
x &gt; distr<SUB><FONT SIZE="-1">out<SUB><FONT SIZE="-1">ggaa<SUB><FONT SIZE="-1">2</FONT></SUB></FONT></SUB></FONT></SUB><BR>
nil &gt; distr<SUB><FONT SIZE="-1">out<SUB><FONT SIZE="-1">ggaa<SUB><FONT SIZE="-1">2</FONT></SUB></FONT></SUB></FONT></SUB><BR>
</BLOCKQUOTE>
<BR>Status: <BLOCKQUOTE>U2<SUB><FONT SIZE="-1">GA<SUB><FONT SIZE="-1">2</FONT></SUB></FONT></SUB>: [1,2]<BR>nil: multiset<BR>U1<SUB><FONT SIZE="-1">ga</FONT></SUB>: []<BR>distr<SUB><FONT SIZE="-1">in<SUB><FONT SIZE="-1">ggaa<SUB><FONT SIZE="-1">2</FONT></SUB></FONT></SUB></FONT></SUB>: multiset<BR>zero: multiset<BR>hbal<SUB><FONT SIZE="-1">tree<SUB><FONT SIZE="-1">in<SUB><FONT SIZE="-1">ga</FONT></SUB></FONT></SUB></FONT></SUB>: []<BR>HBAL<SUB><FONT SIZE="-1">TREE<SUB><FONT SIZE="-1">IN<SUB><FONT SIZE="-1">GA<SUB><FONT SIZE="-1">1</FONT></SUB></FONT></SUB></FONT></SUB></FONT></SUB>: [1]<BR>s<SUB><FONT SIZE="-1">1</FONT></SUB>: multiset<BR>distr<SUB><FONT SIZE="-1">out<SUB><FONT SIZE="-1">ggaa<SUB><FONT SIZE="-1">2</FONT></SUB></FONT></SUB></FONT></SUB>: multiset<BR>hbal<SUB><FONT SIZE="-1">tree<SUB><FONT SIZE="-1">out<SUB><FONT SIZE="-1">ga</FONT></SUB></FONT></SUB></FONT></SUB>: []<BR>x: multiset<BR>U1<SUB><FONT SIZE="-1">GA<SUB><FONT SIZE="-1">1</FONT></SUB></FONT></SUB>: [1]</BLOCKQUOTE><BR><BR>The following usable rules [17] were oriented:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>)
<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>zero</font>)) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>))
<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DL</font>))
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>)
<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DR</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 PiDP</pre><pre>              &#8627 PiDPToQDPProof</pre><pre>                &#8627 QDP</pre><pre>                  &#8627 QDPOrderProof</pre><pre>                    &#8627 <B>QDP</B></pre><pre>                      &#8627 PisEmptyProof</pre><BR>Q DP problem:<BR>P is empty.<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>zero</font>) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>nil</font>)
<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>zero</font>)) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>nil</font>))
<BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>))) &#8594; <FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#0000cc>s</font>(<FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>))
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D1</font>)
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>)
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>D1</font>, <FONT COLOR=#cc0000>D2</font>) &#8594; <FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>D2</font>, <FONT COLOR=#cc0000>D1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#0000cc>distr_out_ggaa</font>(<FONT COLOR=#cc0000>DL</font>, <FONT COLOR=#cc0000>DR</font>)) &#8594; <FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DL</font>))
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>DR</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>DR</font>))
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#cc0000>R</font>)) &#8594; <FONT COLOR=#0000cc>hbal_tree_out_ga</font>(<FONT COLOR=#0000cc>t</font>(<FONT COLOR=#0000cc>x</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>R</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>hbal_tree_in_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>distr_in_ggaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U1_ga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U2_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U3_ga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The TRS P is empty. Hence, there is no (P,Q,R) chain.<BR><BR></body>

