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<title>Left Termination proof of ../tpdb/LP/terminweb/old-terminweb/balance_tree2.pl</title>
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<BR><B>Left Termination</B> of the query pattern
balance_in_2(a, g)
w.r.t. the given <I>Prolog program</I> could not be shown:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>balance(T, TB)&#160;:-&#160;balance(T, -(I, []), -(.(','(TB, -(I, [])), X), X), -(Rest, []), -(Rest, [])).<BR>balance(nil, -(X, X), -(A, B), -(A, B), -(.(','(nil, -(C, C)), T), T)).<BR>balance(tree(L, V, R), -(IH, IT), -(.(','(tree(LB, VB, RB), -(A, D)), H), .(','(LB, -(A, .(VB, X))), .(','(RB, -(X, D)), T))), -(HR, TR), -(NH, NT))&#160;:-&#160;','(balance(L, -(IH, .(V, IT1)), -(H, T), -(HR1, TR1), -(NH, NT1)), balance(R, -(IT1, IT), -(HR1, TR1), -(HR, TR), -(NT1, NT))).<BR><BR>Queries:<BR><BR>balance(a,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>balance_in</font>: (f,b)
<BR><FONT COLOR=#0000cc>balance_in</font>: (f,f,f,f,f)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>)))
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>balance_in_aaaaa</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>balance_in_aaaaa</font><BR>
<FONT COLOR=#0000cc>balance_out_aaaaa</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>balance_out_aaaaa</font><BR>
<FONT COLOR=#0000cc>U2_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>U3_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_out_ag</font><BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 <B>PiTRS</B></pre><pre>      &#8627 DependencyPairsProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>)))
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>balance_in_aaaaa</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>balance_in_aaaaa</font><BR>
<FONT COLOR=#0000cc>balance_out_aaaaa</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>balance_out_aaaaa</font><BR>
<FONT COLOR=#0000cc>U2_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>U3_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_out_ag</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>BALANCE_IN_AG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>BALANCE_IN_AG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))
<BR><FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>U3_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>)))
<BR><FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>)))
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>balance_in_aaaaa</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>balance_in_aaaaa</font><BR>
<FONT COLOR=#0000cc>balance_out_aaaaa</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>balance_out_aaaaa</font><BR>
<FONT COLOR=#0000cc>U2_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>U3_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_out_ag</font><BR>
<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>BALANCE_IN_AAAAA</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>BALANCE_IN_AAAAA</font><BR>
<FONT COLOR=#0000cc>BALANCE_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>BALANCE_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_AAAAA</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AAAAA</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>U2_AAAAA</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 <B>PiDP</B></pre><pre>          &#8627 DependencyGraphProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>BALANCE_IN_AG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>BALANCE_IN_AG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))
<BR><FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>U3_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>)))
<BR><FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>)))
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>balance_in_aaaaa</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>balance_in_aaaaa</font><BR>
<FONT COLOR=#0000cc>balance_out_aaaaa</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>balance_out_aaaaa</font><BR>
<FONT COLOR=#0000cc>U2_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>U3_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_out_ag</font><BR>
<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>BALANCE_IN_AAAAA</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>BALANCE_IN_AAAAA</font><BR>
<FONT COLOR=#0000cc>BALANCE_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>BALANCE_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_AAAAA</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AAAAA</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>U2_AAAAA</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 1 SCC with 3 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 <B>PiDP</B></pre><pre>              &#8627 UsableRulesProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))
<BR><FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>)))
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>balance_in_aaaaa</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>balance_in_aaaaa</font><BR>
<FONT COLOR=#0000cc>balance_out_aaaaa</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>balance_out_aaaaa</font><BR>
<FONT COLOR=#0000cc>U2_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>U3_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_out_ag</font><BR>
<FONT COLOR=#0000cc>BALANCE_IN_AAAAA</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>BALANCE_IN_AAAAA</font><BR>
<FONT COLOR=#0000cc>U2_AAAAA</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>x18</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 PiDP</pre><pre>              &#8627 UsableRulesProof</pre><pre>                &#8627 <B>PiDP</B></pre><pre>                  &#8627 PiDPToQDPProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))
<BR><FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>)))
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>balance_in_aaaaa</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>balance_in_aaaaa</font><BR>
<FONT COLOR=#0000cc>balance_out_aaaaa</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>balance_out_aaaaa</font><BR>
<FONT COLOR=#0000cc>U2_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>U3_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>BALANCE_IN_AAAAA</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>BALANCE_IN_AAAAA</font><BR>
<FONT COLOR=#0000cc>U2_AAAAA</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 PiDP</pre><pre>              &#8627 UsableRulesProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 PiDPToQDPProof</pre><pre>                    &#8627 <B>QDP</B></pre><pre>                      &#8627 Narrowing</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>)
<BR><FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_aaaaa</font> &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font> &#8594; <FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>)
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>) &#8594; <FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>)
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By narrowing [15] the rule <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>) at position [0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>)
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 PiDP</pre><pre>              &#8627 UsableRulesProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 PiDPToQDPProof</pre><pre>                    &#8627 QDP</pre><pre>                      &#8627 Narrowing</pre><pre>                        &#8627 <B>QDP</B></pre><pre>                          &#8627 NonTerminationProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>)
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>))
<BR><FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_aaaaa</font> &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font> &#8594; <FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>)
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>) &#8594; <FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>)
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We used the non-termination processor [17] to show that the DP problem is infinite.<BR>Found a loop by semiunifying a rule from P directly.<BR><BR>The TRS P consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>)
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>))
<BR><FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_aaaaa</font> &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font> &#8594; <FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>)
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>) &#8594; <FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>)
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font></BLOCKQUOTE><BR><BR>s = <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> evaluates to  t =<FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font><BR><BR>Thus s starts an infinite chain as s semiunifies with t with the following substitutions:<BR><UL><LI> Semiunifier: [ ]</LI>
<LI> Matcher: [ ]</LI></UL><BR><BR><hr><BR><B>Rewriting sequence</B><BR><BR>The DP semiunifies directly so there is only one rewrite step from BALANCE_IN_AAAAA to BALANCE_IN_AAAAA.<BR><BR><BR><BR><BR>We use the technique of [30]. With regard to the inferred argument filtering the predicates were used in the following modes:
<BR><FONT COLOR=#0000cc>balance_in</font>: (f,b)
<BR><FONT COLOR=#0000cc>balance_in</font>: (f,f,f,f,f)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>)))
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>balance_in_aaaaa</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>balance_in_aaaaa</font><BR>
<FONT COLOR=#0000cc>balance_out_aaaaa</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>balance_out_aaaaa</font><BR>
<FONT COLOR=#0000cc>U2_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>U3_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 <B>PiTRS</B></pre><pre>      &#8627 DependencyPairsProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>)))
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>balance_in_aaaaa</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>balance_in_aaaaa</font><BR>
<FONT COLOR=#0000cc>balance_out_aaaaa</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>balance_out_aaaaa</font><BR>
<FONT COLOR=#0000cc>U2_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>U3_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>x2</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>BALANCE_IN_AG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>BALANCE_IN_AG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))
<BR><FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>U3_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>)))
<BR><FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>)))
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>balance_in_aaaaa</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>balance_in_aaaaa</font><BR>
<FONT COLOR=#0000cc>balance_out_aaaaa</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>balance_out_aaaaa</font><BR>
<FONT COLOR=#0000cc>U2_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>U3_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>BALANCE_IN_AAAAA</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>BALANCE_IN_AAAAA</font><BR>
<FONT COLOR=#0000cc>BALANCE_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>BALANCE_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_AAAAA</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AAAAA</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>U2_AAAAA</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 <B>PiDP</B></pre><pre>          &#8627 DependencyGraphProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>BALANCE_IN_AG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>) &#8594; <FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>BALANCE_IN_AG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>))
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))
<BR><FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>U3_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>)))
<BR><FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>)))
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>balance_in_aaaaa</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>balance_in_aaaaa</font><BR>
<FONT COLOR=#0000cc>balance_out_aaaaa</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>balance_out_aaaaa</font><BR>
<FONT COLOR=#0000cc>U2_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>U3_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_AG</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>BALANCE_IN_AAAAA</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>BALANCE_IN_AAAAA</font><BR>
<FONT COLOR=#0000cc>BALANCE_IN_AG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>BALANCE_IN_AG</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_AAAAA</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_AAAAA</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>U2_AAAAA</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 1 SCC with 3 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 <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>BALANCE_IN_AAAAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))
<BR><FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>) &#8594; <FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>)))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>)))
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>))
<BR><FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>TB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>I</font>, <FONT COLOR=#0000cc>[]</font>)), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>Rest</font>, <FONT COLOR=#0000cc>[]</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>TB</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_in_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_ag</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>balance_in_aaaaa</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>balance_in_aaaaa</font><BR>
<FONT COLOR=#0000cc>balance_out_aaaaa</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>balance_out_aaaaa</font><BR>
<FONT COLOR=#0000cc>U2_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>U3_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>balance_out_ag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>BALANCE_IN_AAAAA</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>BALANCE_IN_AAAAA</font><BR>
<FONT COLOR=#0000cc>U2_AAAAA</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 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>BALANCE_IN_AAAAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))
<BR><FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>B</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>nil</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>C</font>, <FONT COLOR=#cc0000>C</font>)), <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>)) &#8594; <FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>)))
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>IT1</font>)), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT1</font>))) &#8594; <FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_in_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>)))
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>, <FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>, <FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>, <FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>, <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IT1</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR1</font>, <FONT COLOR=#cc0000>TR1</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NT1</font>, <FONT COLOR=#cc0000>NT</font>))) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>V</font>, <FONT COLOR=#cc0000>R</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>IH</font>, <FONT COLOR=#cc0000>IT</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#0000cc>tree</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>RB</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>H</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>LB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>A</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>VB</font>, <FONT COLOR=#cc0000>X</font>))), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#0000cc>','</font>(<FONT COLOR=#cc0000>RB</font>, <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>D</font>)), <FONT COLOR=#cc0000>T</font>))), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>HR</font>, <FONT COLOR=#cc0000>TR</font>), <FONT COLOR=#0000cc>-</font>(<FONT COLOR=#cc0000>NH</font>, <FONT COLOR=#cc0000>NT</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>balance_in_aaaaa</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>balance_in_aaaaa</font><BR>
<FONT COLOR=#0000cc>balance_out_aaaaa</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>balance_out_aaaaa</font><BR>
<FONT COLOR=#0000cc>U2_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>U3_aaaaa</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<FONT COLOR=#0000cc>BALANCE_IN_AAAAA</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>BALANCE_IN_AAAAA</font><BR>
<FONT COLOR=#0000cc>U2_AAAAA</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>, <FONT COLOR=#cc0000>x7</font>, <FONT COLOR=#cc0000>x8</font>, <FONT COLOR=#cc0000>x9</font>, <FONT COLOR=#cc0000>x10</font>, <FONT COLOR=#cc0000>x11</font>, <FONT COLOR=#cc0000>x12</font>, <FONT COLOR=#cc0000>x13</font>, <FONT COLOR=#cc0000>x14</font>, <FONT COLOR=#cc0000>x15</font>, <FONT COLOR=#cc0000>x16</font>, <FONT COLOR=#cc0000>x17</font>, <FONT COLOR=#cc0000>x18</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#cc0000>x18</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 PiDP</pre><pre>              &#8627 UsableRulesProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 PiDPToQDPProof</pre><pre>                    &#8627 <B>QDP</B></pre><pre>                      &#8627 Narrowing</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>)
<BR><FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_aaaaa</font> &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font> &#8594; <FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>)
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>) &#8594; <FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>)
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By narrowing [15] the rule <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>) at position [0] we obtained the following new rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>)
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>))</BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 PiDP</pre><pre>              &#8627 UsableRulesProof</pre><pre>                &#8627 PiDP</pre><pre>                  &#8627 PiDPToQDPProof</pre><pre>                    &#8627 QDP</pre><pre>                      &#8627 Narrowing</pre><pre>                        &#8627 <B>QDP</B></pre><pre>                          &#8627 NonTerminationProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>)
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>))
<BR><FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_aaaaa</font> &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font> &#8594; <FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>)
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>) &#8594; <FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>)
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font></BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_aaaaa</font>
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#cc0000>x0</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>We used the non-termination processor [17] to show that the DP problem is infinite.<BR>Found a loop by semiunifying a rule from P directly.<BR><BR>The TRS P consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>)
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>))
<BR><FONT COLOR=#0000cc>U2_AAAAA</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>) &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font>
<BR><FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> &#8594; <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font></BLOCKQUOTE><BR>The TRS R consists of the following rules:<BLOCKQUOTE><BR><FONT COLOR=#0000cc>balance_in_aaaaa</font> &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font>
<BR><FONT COLOR=#0000cc>balance_in_aaaaa</font> &#8594; <FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>)
<BR><FONT COLOR=#0000cc>U2_aaaaa</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>) &#8594; <FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#0000cc>balance_in_aaaaa</font>)
<BR><FONT COLOR=#0000cc>U3_aaaaa</font>(<FONT COLOR=#0000cc>balance_out_aaaaa</font>) &#8594; <FONT COLOR=#0000cc>balance_out_aaaaa</font></BLOCKQUOTE><BR><BR>s = <FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font> evaluates to  t =<FONT COLOR=#0000cc>BALANCE_IN_AAAAA</font><BR><BR>Thus s starts an infinite chain as s semiunifies with t with the following substitutions:<BR><UL><LI> Semiunifier: [ ]</LI>
<LI> Matcher: [ ]</LI></UL><BR><BR><hr><BR><B>Rewriting sequence</B><BR><BR>The DP semiunifies directly so there is only one rewrite step from BALANCE_IN_AAAAA to BALANCE_IN_AAAAA.<BR><BR><BR><BR><BR><BR></body>

