YES
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<title>Left Termination proof of ../tpdb/LP/talp/taboch/sameleaves.pl</title>
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<BR><B>Left Termination</B> of the query pattern
sameleaves_in_2(g, g)
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>sameleaves(leaf(L), leaf(L)).<BR>sameleaves(tree(T1, T2), tree(S1, S2))&#160;:-&#160;','(getleave(T1, T2, L, T), ','(getleave(S1, S2, L, S), sameleaves(T, S))).<BR>getleave(leaf(A), C, A, C).<BR>getleave(tree(A, B), C, L, O)&#160;:-&#160;getleave(A, tree(B, C), L, O).<BR><BR>Queries:<BR><BR>sameleaves(g,g).<BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>sameleaves_in</font>: (b,b)
<BR><FONT COLOR=#0000cc>getleave_in</font>: (b,b,f,f) (b,b,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>sameleaves_in_gg</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>), <FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>), <FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>)) &#8594; <FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>sameleaves_out_gg</font><BR>
<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>getleave_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>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>getleave_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>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ggaa</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_ggaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_gg</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_gg</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>getleave_in_ggga</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>getleave_in_ggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>getleave_out_ggga</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>getleave_out_ggga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ggga</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_ggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x5</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>sameleaves_in_gg</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>), <FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>), <FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>)) &#8594; <FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>sameleaves_out_gg</font><BR>
<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>getleave_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>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>getleave_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>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ggaa</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_ggaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_gg</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_gg</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>getleave_in_ggga</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>getleave_in_ggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>getleave_out_ggga</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>getleave_out_ggga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ggga</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_ggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x5</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>SAMELEAVES_IN_GG</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>)) &#8594; <FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>)) &#8594; <FONT COLOR=#0000cc>GETLEAVE_IN_GGAA</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>GETLEAVE_IN_GGAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_GGAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>GETLEAVE_IN_GGAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>GETLEAVE_IN_GGAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>U3_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>), <FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>), <FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>)) &#8594; <FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>sameleaves_out_gg</font><BR>
<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>getleave_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>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>getleave_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>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ggaa</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_ggaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_gg</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_gg</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>getleave_in_ggga</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>getleave_in_ggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>getleave_out_ggga</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>getleave_out_ggga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ggga</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_ggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>GETLEAVE_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>GETLEAVE_IN_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_GGAA</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_GGAA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>GETLEAVE_IN_GGGA</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>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_GG</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_GG</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_GGGA</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_GGGA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GG</font>(<FONT COLOR=#cc0000>x5</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>SAMELEAVES_IN_GG</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>)) &#8594; <FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>)) &#8594; <FONT COLOR=#0000cc>GETLEAVE_IN_GGAA</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)
<BR><FONT COLOR=#0000cc>GETLEAVE_IN_GGAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_GGAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>GETLEAVE_IN_GGAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>GETLEAVE_IN_GGAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_GGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>U3_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>), <FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>), <FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>)) &#8594; <FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>sameleaves_out_gg</font><BR>
<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>getleave_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>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>getleave_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>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ggaa</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_ggaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_gg</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_gg</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>getleave_in_ggga</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>getleave_in_ggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>getleave_out_ggga</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>getleave_out_ggga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ggga</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_ggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>GETLEAVE_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>GETLEAVE_IN_GGAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U4_GGAA</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_GGAA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>GETLEAVE_IN_GGGA</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>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U2_GG</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_GG</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U4_GGGA</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_GGGA</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_GG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 3 SCCs with 5 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><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>), <FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>), <FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>)) &#8594; <FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>sameleaves_out_gg</font><BR>
<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>getleave_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>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>getleave_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>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ggaa</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_ggaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_gg</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_gg</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>getleave_in_ggga</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>getleave_in_ggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>getleave_out_ggga</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>getleave_out_ggga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ggga</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_ggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>GETLEAVE_IN_GGGA</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>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>GETLEAVE_IN_GGGA</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>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</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><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</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>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>GETLEAVE_IN_GGGA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>)<BR>The graph contains the following edges 1 > 1, 3 >= 3<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><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>GETLEAVE_IN_GGAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>GETLEAVE_IN_GGAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>), <FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>), <FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>)) &#8594; <FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>sameleaves_out_gg</font><BR>
<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>getleave_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>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>getleave_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>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ggaa</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_ggaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_gg</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_gg</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>getleave_in_ggga</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>getleave_in_ggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>getleave_out_ggga</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>getleave_out_ggga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ggga</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_ggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>GETLEAVE_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>GETLEAVE_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>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><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>GETLEAVE_IN_GGAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>GETLEAVE_IN_GGAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>GETLEAVE_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>GETLEAVE_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>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><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>GETLEAVE_IN_GGAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>GETLEAVE_IN_GGAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</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>GETLEAVE_IN_GGAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>GETLEAVE_IN_GGAA</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</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><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>)) &#8594; <FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>), <FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>)) &#8594; <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>), <FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>)) &#8594; <FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U2_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>sameleaves_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>sameleaves_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>sameleaves_out_gg</font><BR>
<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_gg</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>getleave_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>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>getleave_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>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ggaa</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_ggaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U2_gg</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_gg</font>(<FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>getleave_in_ggga</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>getleave_in_ggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>getleave_out_ggga</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>getleave_out_ggga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ggga</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_ggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_gg</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_GG</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_GG</font>(<FONT COLOR=#cc0000>x5</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><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>))
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>)
<BR><FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>)) &#8594; <FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>) &#8594; <FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>))
<BR><FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>getleave_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>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>getleave_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>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ggaa</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_ggaa</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>getleave_in_ggga</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>getleave_in_ggga</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>getleave_out_ggga</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>getleave_out_ggga</font>(<FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>U4_ggga</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_ggga</font>(<FONT COLOR=#cc0000>x6</font>)<BR>
<FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U2_GG</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_GG</font>(<FONT COLOR=#cc0000>x5</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 UsableRulesReductionPairsProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>)) &#8594; <FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.<BR><BR>No dependency pairs are removed.<BR><BR>The following rules are removed from R:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>A</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>C</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>leaf</font>(<FONT COLOR=#cc0000>A</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>C</font>)</BLOCKQUOTE>Used ordering: POLO with Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 2&middot;x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_GG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = x<SUB>1</SUB> + x<SUB>2</SUB> + x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_GG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 2&middot;x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_ggaa</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_ggga</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>getleave_in_ggaa</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 2&middot;x<SUB>1</SUB> + 2&middot;x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>getleave_in_ggga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = x<SUB>1</SUB> + x<SUB>2</SUB> + 2&middot;x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>getleave_out_ggaa</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 2&middot;x<SUB>1</SUB> + 2&middot;x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>getleave_out_ggga</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>leaf</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>tree</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 UsableRulesReductionPairsProof</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 RuleRemovalProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>)) &#8594; <FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
<BR>Strictly oriented dependency pairs:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#cc0000>L</font>))</BLOCKQUOTE><BR><BR>Used ordering: POLO with Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 2&middot;x<SUB>1</SUB> + 2&middot;x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U1_GG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 2&middot;x<SUB>1</SUB> + 2&middot;x<SUB>2</SUB> + x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U2_GG</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 2&middot;x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_ggaa</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U4_ggga</font>(x<SUB>1</SUB>)</B>) = x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>getleave_in_ggaa</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 2&middot;x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>getleave_in_ggga</font>(x<SUB>1</SUB>, x<SUB>2</SUB>, x<SUB>3</SUB>)</B>) = 2&middot;x<SUB>1</SUB> + 2&middot;x<SUB>2</SUB> + x<SUB>3</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>getleave_out_ggaa</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + 2&middot;x<SUB>1</SUB> + 2&middot;x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>getleave_out_ggga</font>(x<SUB>1</SUB>)</B>) = 2&middot;x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>tree</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 UsableRulesReductionPairsProof</pre><pre>                          &#8627 QDP</pre><pre>                            &#8627 RuleRemovalProof</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>SAMELEAVES_IN_GG</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>), <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>)) &#8594; <FONT COLOR=#0000cc>U1_GG</font>(<FONT COLOR=#cc0000>S1</font>, <FONT COLOR=#cc0000>S2</font>, <FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>T1</font>, <FONT COLOR=#cc0000>T2</font>))
<BR><FONT COLOR=#0000cc>U2_GG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>S</font>)) &#8594; <FONT COLOR=#0000cc>SAMELEAVES_IN_GG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>S</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>), <FONT COLOR=#cc0000>L</font>))
<BR><FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggga</font>(<FONT COLOR=#cc0000>O</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#cc0000>C</font>) &#8594; <FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>B</font>, <FONT COLOR=#cc0000>C</font>)))
<BR><FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)) &#8594; <FONT COLOR=#0000cc>getleave_out_ggaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>O</font>)</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>getleave_in_ggga</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)
<BR><FONT COLOR=#0000cc>getleave_in_ggaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)
<BR><FONT COLOR=#0000cc>U4_ggga</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U4_ggaa</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 2 less nodes.<BR><BR></body>

