YES
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Frameset//EN"
"http:/www.w3.org/TR/html4/frameset.dtd">
<html>
<head>
<title>Left Termination proof of ../tpdb/LP/SGST06/palindrome.pl</title>
</head>
<body>
<BR><B>Left Termination</B> of the query pattern
palindrome_in_1(g)
w.r.t. the given <I>Prolog program</I> could successfully be <font color=#00ff00>proven</font>:<BR><BR><BR><BR><pre>&#8627 <B>Prolog</B></pre><pre>  &#8627 PrologToPiTRSProof</pre><BR>Clauses:<BR><BR>palindrome(L)&#160;:-&#160;','(halves(L, X1s, X2s, EvenOdd), ','(eq(EvenOdd, even), eq(X1s, X2s))).<BR>palindrome(L)&#160;:-&#160;','(halves(L, X1s, X2s, EvenOdd), ','(eq(EvenOdd, odd), last(X1s, X, X2s))).<BR>halves([], [], [], even).<BR>halves(.(X, []), .(X, []), [], odd).<BR>halves(.(T, .(Y, Xs)), .(T, Ts), .(R, Rs), EvenOdd)&#160;:-&#160;','(last(.(Y, Xs), R, Rests), halves(Rests, Ts, Rs, EvenOdd)).<BR>last(.(T, []), T, []).<BR>last(.(H, T), X, .(H, M))&#160;:-&#160;last(T, X, M).<BR>eq(X, X).<BR><BR>Queries:<BR><BR>palindrome(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>palindrome_in</font>: (b)
<BR><FONT COLOR=#0000cc>halves_in</font>: (b,f,f,f)
<BR><FONT COLOR=#0000cc>last_in</font>: (b,f,f) (b,f,b)
<BR>Transforming <I>Prolog</I> into the following <B>Term Rewriting System</B>:
<BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>))
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>even</font>) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>even</font>)
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>odd</font>) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>odd</font>)
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>) &#8594; <FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>))
<BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>)) &#8594; <FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>))
<BR><FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>)
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>))
<BR><FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>)) &#8594; <FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>))
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>)) &#8594; <FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U8_gag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U8_gag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gaaa</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_gaaa</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_gg</font><BR>
<FONT COLOR=#0000cc>even</font>&#160; = &#160;<FONT COLOR=#0000cc>even</font><BR>
<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>palindrome_out_g</font><BR>
<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>odd</font>&#160; = &#160;<FONT COLOR=#0000cc>odd</font><BR>
<FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_gag</font>(<FONT 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>U8_gag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<BR>
<P><B>Infinitary Constructor Rewriting Termination</B> of PiTRS implies <B>Termination</B> of Prolog<P>
<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 <B>PiTRS</B></pre><pre>      &#8627 DependencyPairsProof</pre><BR>Pi-finite rewrite system:<BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>))
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>even</font>) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>even</font>)
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>odd</font>) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>odd</font>)
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>) &#8594; <FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>))
<BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>)) &#8594; <FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>))
<BR><FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>)
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>))
<BR><FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>)) &#8594; <FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>))
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>)) &#8594; <FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U8_gag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U8_gag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gaaa</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_gaaa</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_gg</font><BR>
<FONT COLOR=#0000cc>even</font>&#160; = &#160;<FONT COLOR=#0000cc>even</font><BR>
<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>palindrome_out_g</font><BR>
<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>odd</font>&#160; = &#160;<FONT COLOR=#0000cc>odd</font><BR>
<FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_gag</font>(<FONT 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>U8_gag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<BR><BR>Using Dependency Pairs [1,30] we result in the following initial DP problem:<BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>PALINDROME_IN_G</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>))
<BR><FONT COLOR=#0000cc>PALINDROME_IN_G</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)
<BR><FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>) &#8594; <FONT COLOR=#0000cc>U6_GAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>))
<BR><FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>) &#8594; <FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>)
<BR><FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U8_GAA</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)
<BR><FONT COLOR=#0000cc>U6_GAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>)) &#8594; <FONT COLOR=#0000cc>U7_GAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>))
<BR><FONT COLOR=#0000cc>U6_GAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>)) &#8594; <FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>)
<BR><FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>))
<BR><FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>EQ_IN_GG</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>)
<BR><FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>)) &#8594; <FONT COLOR=#0000cc>U3_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>)) &#8594; <FONT COLOR=#0000cc>EQ_IN_GG</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)
<BR><FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>U4_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>))
<BR><FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>EQ_IN_GG</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>)
<BR><FONT COLOR=#0000cc>U4_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>)) &#8594; <FONT COLOR=#0000cc>U5_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U4_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>)
<BR><FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U8_GAG</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>))
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>even</font>) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>even</font>)
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>odd</font>) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>odd</font>)
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>) &#8594; <FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>))
<BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>)) &#8594; <FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>))
<BR><FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>)
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>))
<BR><FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>)) &#8594; <FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>))
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>)) &#8594; <FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U8_gag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U8_gag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gaaa</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_gaaa</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_gg</font><BR>
<FONT COLOR=#0000cc>even</font>&#160; = &#160;<FONT COLOR=#0000cc>even</font><BR>
<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>palindrome_out_g</font><BR>
<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>odd</font>&#160; = &#160;<FONT COLOR=#0000cc>odd</font><BR>
<FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_gag</font>(<FONT 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>U8_gag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_GAAA</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_G</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_GAG</font>(<FONT 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>U8_GAG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>EQ_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>EQ_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_GAAA</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<FONT COLOR=#0000cc>U4_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_G</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>PALINDROME_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>PALINDROME_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_G</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 <B>PiDP</B></pre><pre>          &#8627 DependencyGraphProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>PALINDROME_IN_G</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>))
<BR><FONT COLOR=#0000cc>PALINDROME_IN_G</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)
<BR><FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>) &#8594; <FONT COLOR=#0000cc>U6_GAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>))
<BR><FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>) &#8594; <FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>)
<BR><FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U8_GAA</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)
<BR><FONT COLOR=#0000cc>U6_GAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>)) &#8594; <FONT COLOR=#0000cc>U7_GAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>))
<BR><FONT COLOR=#0000cc>U6_GAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>)) &#8594; <FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>)
<BR><FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>))
<BR><FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>EQ_IN_GG</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>)
<BR><FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>)) &#8594; <FONT COLOR=#0000cc>U3_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>)) &#8594; <FONT COLOR=#0000cc>EQ_IN_GG</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)
<BR><FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>U4_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>))
<BR><FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>EQ_IN_GG</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>)
<BR><FONT COLOR=#0000cc>U4_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>)) &#8594; <FONT COLOR=#0000cc>U5_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U4_G</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>)
<BR><FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U8_GAG</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>))
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>even</font>) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>even</font>)
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>odd</font>) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>odd</font>)
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>) &#8594; <FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>))
<BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>)) &#8594; <FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>))
<BR><FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>)
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>))
<BR><FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>)) &#8594; <FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>))
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>)) &#8594; <FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U8_gag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U8_gag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gaaa</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_gaaa</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_gg</font><BR>
<FONT COLOR=#0000cc>even</font>&#160; = &#160;<FONT COLOR=#0000cc>even</font><BR>
<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>palindrome_out_g</font><BR>
<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>odd</font>&#160; = &#160;<FONT COLOR=#0000cc>odd</font><BR>
<FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_gag</font>(<FONT 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>U8_gag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_GAAA</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_GAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_G</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U3_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_G</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_GAG</font>(<FONT 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>U8_GAG</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>EQ_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>EQ_IN_GG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U6_GAAA</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<FONT COLOR=#0000cc>U4_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_G</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>PALINDROME_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>PALINDROME_IN_G</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U5_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_G</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_G</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>The approximation of the Dependency Graph [30] contains 3 SCCs with 14 less nodes.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>))
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>even</font>) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>even</font>)
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>odd</font>) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>odd</font>)
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>) &#8594; <FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>))
<BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>)) &#8594; <FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>))
<BR><FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>)
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>))
<BR><FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>)) &#8594; <FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>))
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>)) &#8594; <FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U8_gag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U8_gag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gaaa</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_gaaa</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_gg</font><BR>
<FONT COLOR=#0000cc>even</font>&#160; = &#160;<FONT COLOR=#0000cc>even</font><BR>
<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>palindrome_out_g</font><BR>
<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>odd</font>&#160; = &#160;<FONT COLOR=#0000cc>odd</font><BR>
<FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_gag</font>(<FONT 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>U8_gag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_GAG</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>M</font>)<BR>The graph contains the following edges 1 > 1, 2 > 2<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>))
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>even</font>) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>even</font>)
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>odd</font>) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>odd</font>)
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>) &#8594; <FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>))
<BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>)) &#8594; <FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>))
<BR><FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>)
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>))
<BR><FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>)) &#8594; <FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>))
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>)) &#8594; <FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U8_gag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U8_gag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gaaa</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_gaaa</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_gg</font><BR>
<FONT COLOR=#0000cc>even</font>&#160; = &#160;<FONT COLOR=#0000cc>even</font><BR>
<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>palindrome_out_g</font><BR>
<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>odd</font>&#160; = &#160;<FONT COLOR=#0000cc>odd</font><BR>
<FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_gag</font>(<FONT 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>U8_gag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>              &#8627 PiDP</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)</BLOCKQUOTE><BR>R is empty.<BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 QDPSizeChangeProof</pre><pre>              &#8627 PiDP</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#cc0000>T</font>)</BLOCKQUOTE><BR>R is empty.<BR>Q is empty.<BR>We have to consider all (P,Q,R)-chains.<BR>By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem. <P>From the DPs we obtained the following set of size-change graphs:
<UL><LI><FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>LAST_IN_GAA</font>(<FONT COLOR=#cc0000>T</font>)<BR>The graph contains the following edges 1 > 1<P></LI></UL><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 <B>PiDP</B></pre><pre>                &#8627 UsableRulesProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U6_GAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>)) &#8594; <FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>)
<BR><FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>) &#8594; <FONT COLOR=#0000cc>U6_GAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>L</font>) &#8594; <FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>))
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>even</font>) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>even</font>)
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>odd</font>) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#0000cc>[]</font>, <FONT COLOR=#0000cc>odd</font>)
<BR><FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>) &#8594; <FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>))
<BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>)) &#8594; <FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>))
<BR><FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>)
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>))
<BR><FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>) &#8594; <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X</font>)
<BR><FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>even</font>)) &#8594; <FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>L</font>)
<BR><FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#cc0000>EvenOdd</font>)) &#8594; <FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>))
<BR><FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X2s</font>, <FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>odd</font>)) &#8594; <FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>))
<BR><FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U8_gag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U8_gag</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>L</font>, <FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>X1s</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>X2s</font>)) &#8594; <FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>L</font>)</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>palindrome_in_g</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U1_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>halves_in_gaaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>halves_out_gaaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U6_gaaa</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U6_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>U7_gaaa</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U7_gaaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U2_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_in_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>eq_out_gg</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>eq_out_gg</font><BR>
<FONT COLOR=#0000cc>even</font>&#160; = &#160;<FONT COLOR=#0000cc>even</font><BR>
<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U3_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>palindrome_out_g</font>(<FONT COLOR=#cc0000>x1</font>)&#160; = &#160;<FONT COLOR=#0000cc>palindrome_out_g</font><BR>
<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>U4_g</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)<BR>
<FONT COLOR=#0000cc>odd</font>&#160; = &#160;<FONT COLOR=#0000cc>odd</font><BR>
<FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>U5_g</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_gag</font>(<FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>U8_gag</font>(<FONT 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>U8_gag</font>(<FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_GAAA</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 <B>PiDP</B></pre><pre>                    &#8627 PiDPToQDPProof</pre><BR>Pi DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>U6_GAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>)) &#8594; <FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#cc0000>Rests</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>)
<BR><FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Ts</font>), <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>), <FONT COLOR=#cc0000>EvenOdd</font>) &#8594; <FONT COLOR=#0000cc>U6_GAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>, <FONT COLOR=#cc0000>Ts</font>, <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rs</font>, <FONT COLOR=#cc0000>EvenOdd</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>), <FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>))</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>), <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>))
<BR><FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>), <FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>))</BLOCKQUOTE><BR>The argument filtering Pi contains the following mapping:<BR><FONT COLOR=#0000cc>[]</font>&#160; = &#160;<FONT COLOR=#0000cc>[]</font><BR>
<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)&#160; = &#160;<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>)<BR>
<FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)&#160; = &#160;<FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>)<BR>
<FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>, <FONT COLOR=#cc0000>x5</font>)&#160; = &#160;<FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x5</font>)<BR>
<FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x2</font>, <FONT COLOR=#cc0000>x3</font>, <FONT COLOR=#cc0000>x4</font>)&#160; = &#160;<FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#cc0000>x1</font>)<BR>
<FONT COLOR=#0000cc>U6_GAAA</font>(<FONT 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>)&#160; = &#160;<FONT COLOR=#0000cc>U6_GAAA</font>(<FONT COLOR=#cc0000>x1</font>, <FONT COLOR=#cc0000>x8</font>)<BR>
<BR>We have to consider all (P,R,Pi)-chains<BR>Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.<BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 <B>QDP</B></pre><pre>                        &#8627 RuleRemovalProof</pre><BR>Q DP problem:<BR>The TRS P consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U6_GAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U6_GAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>)) &#8594; <FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#cc0000>Rests</font>)</BLOCKQUOTE><BR>The TRS R consists of the following rules:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>))</BLOCKQUOTE><BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
<BR>Strictly oriented dependency pairs:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>))) &#8594; <FONT COLOR=#0000cc>U6_GAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>Y</font>, <FONT COLOR=#cc0000>Xs</font>)))
<BR><FONT COLOR=#0000cc>U6_GAAA</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>R</font>, <FONT COLOR=#cc0000>Rests</font>)) &#8594; <FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(<FONT COLOR=#cc0000>Rests</font>)</BLOCKQUOTE><BR>Strictly oriented rules of the TRS R:
<BLOCKQUOTE><BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>)) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>T</font>, <FONT COLOR=#0000cc>[]</font>)
<BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>T</font>)) &#8594; <FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>T</font>))
<BR><FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#cc0000>M</font>)) &#8594; <FONT COLOR=#0000cc>last_out_gaa</font>(<FONT COLOR=#cc0000>X</font>, <FONT COLOR=#0000cc>.</font>(<FONT COLOR=#cc0000>H</font>, <FONT COLOR=#cc0000>M</font>))</BLOCKQUOTE><BR>Used ordering: POLO with Polynomial interpretation [25]:
<BLOCKQUOTE><BR>POL(<B><FONT COLOR=#0000cc>.</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 2 + 2&middot;x<SUB>1</SUB> + 2&middot;x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>HALVES_IN_GAAA</font>(x<SUB>1</SUB>)</B>) = 1 + 2&middot;x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U6_GAAA</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 2 + 2&middot;x<SUB>1</SUB> + 2&middot;x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>U8_gaa</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 1 + 2&middot;x<SUB>1</SUB> + 2&middot;x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>[]</font></B>) = 1<sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>last_in_gaa</font>(x<SUB>1</SUB>)</B>) = 2&middot;x<SUB>1</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub>
<BR>POL(<B><FONT COLOR=#0000cc>last_out_gaa</font>(x<SUB>1</SUB>, x<SUB>2</SUB>)</B>) = 2 + x<SUB>1</SUB> + x<SUB>2</SUB><sup>&nbsp;</sup> <sub>&nbsp;</sub></BLOCKQUOTE><BR><BR><BR><pre>&#8627 Prolog</pre><pre>  &#8627 PrologToPiTRSProof</pre><pre>    &#8627 PiTRS</pre><pre>      &#8627 DependencyPairsProof</pre><pre>        &#8627 PiDP</pre><pre>          &#8627 DependencyGraphProof</pre><pre>            &#8627 AND</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>              &#8627 PiDP</pre><pre>                &#8627 UsableRulesProof</pre><pre>                  &#8627 PiDP</pre><pre>                    &#8627 PiDPToQDPProof</pre><pre>                      &#8627 QDP</pre><pre>                        &#8627 RuleRemovalProof</pre><pre>                          &#8627 <B>QDP</B></pre><pre>                            &#8627 PisEmptyProof</pre><BR>Q DP problem:<BR>P is empty.<BR>R is empty.<BR>The set Q consists of the following terms:<BR><BLOCKQUOTE><BR><FONT COLOR=#0000cc>last_in_gaa</font>(<FONT COLOR=#cc0000>x0</font>)
<BR><FONT COLOR=#0000cc>U8_gaa</font>(<FONT COLOR=#cc0000>x0</font>, <FONT COLOR=#cc0000>x1</font>)</BLOCKQUOTE><BR>We have to consider all (P,Q,R)-chains.<BR>The TRS P is empty. Hence, there is no (P,Q,R) chain.<BR><BR></body>

